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Introduction to the Theory of Hyperbolic Conservation Laws
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Introduction to the Theory of Hyperbolic Conservation Laws C.M. Dafermos Division of Applied Mathematics, Brown University, Providence, RI, United States
Chapter Outline 1 Introduction 2 Basic Structure of Hyperbolic Conservation Laws
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3 Strictly Hyperbolic Systems in One Spatial Dimension References
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ABSTRACT This is a brief, informal introduction to nonlinear hyperbolic conservation laws, underscoring their inherent properties (wave breaking, entropy conditions) and sketching the state of the art in their analysis. Keywords: Hyperbolic conservation laws, Entropy, Viscosity, Shocks, Riemann Problem AMS Classification Codes: 35L65, 35L67
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INTRODUCTION
The conservation laws of gas dynamics, elastodynamics, electrodynamics and other branches of classical physics are typically expressed by hyperbolic partial differential equations or systems thereof. In particular, it is hyperbolic systems that provide the proper mathematical setting for a host of wave phenomena. The salient feature of solutions to nonlinear hyperbolic systems resulting from conservation laws is wave breaking, which triggers the development of jump discontinuities that propagate on as shock waves. This renders the mathematical theory particularly hard, as it must cope with weak solutions. The difficulty is exacerbated by the fact that uniqueness and stability are lost in the realm of weak solutions. As a remedy, one seeks selection criteria, motivated by physical or mathematical considerations, that hopefully weed out all spurious solutions, singling out the admissible one. In consequence of these difficulties, and despite considerable progress achieved over the past 50 years, Handbook of Numerical Analysis. http://dx.doi.org/10.1016/bs.hna.2016.08.003 © 2016 Elsevier B.V. All rights reserved.
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central mathematical issues are still wide open, especially in several spatial dimensions. It is fair to admit that at the present time computation has outpaced the theory. The aim of this introductory chapter is to provide a description of the basic structure of hyperbolic systems of conservation laws and to survey their inherent properties. We shall not provide specific bibliographic references. The reader who seeks more detailed information should consult one or more of numerous existing texts. The list includes Smoller (1994) (a clear introduction to the basics, albeit somewhat dated), Holden and Risebro (2015) (a readable, successful marriage of theory with numerics), Serre (1999) (an insightful introduction to the basics, supplemented with an interesting selection of more advanced topics), Bressan (2000) (a nice exposition starting out at an introductory elementary level and becoming progressively more focused and technical) and Dafermos (2016) (an encyclopaedic coverage of the field, with voluminous bibliography). The perspective and style of presentation in this chapter are borrowed from Dafermos (2016). The author is indebted to Zheng Sun for his valuable assistance in drawing the figures.
2 BASIC STRUCTURE OF HYPERBOLIC CONSERVATION LAWS The canonical form of a system of n conservation laws in k spatial dimensions reads k X (1) @t U + @a Fa ðUÞ ¼ 0: a¼1
The (unknown) n-dimensional state vector field U is a function of the k-dimensional spatial variable x and the scalar temporal variable t. For a ¼ 1,…, k, the flux Fa(U) is a given smooth function from n to n and @ a stands for @/@xa. The terminology, with origins in classical physics, stems from the observation that (1) holds on some domain of k if and only if Z I X k d (2) Udx + a Fa ðUÞdS ¼ 0 dt O @O a¼1 for all smooth subdomains O, with @O denoting the boundary of O and standing for the exterior unit normal on @O. Indeed, the ‘physical’ interpretation of (2) is that the n-vector valued quantity with density U is conserved, in the sense that the rate of change in the amount stored in O is balanced by the rate of flux, in or out of O, through @O. In what follows we will employ matrix notation, identifying n with column vectors n1 . The symbol D will denote the gradient operator in n , mapping scalar fields into 1n row vector fields and n1 column vector fields into nn matrix fields.
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The system of conservation laws (1) will Pbe called hyperbolic if for any fixed U 2 n and 2 k1 , the n n matrix ka¼1 a DFa ðUÞ has real eigenvalues l1(U, ),…, ln(U, ) and an associated set of n linearly independent eigenvectors R1(U, ),…, Rn(U, ). An important subclass of hyperbolic conservation laws are the symmetric systems (1) with the property that for a ¼ 1,…, k and any U 2 n , the matrices DFa(U) are symmetric. As a consequence of hyperbolicity, for any fixed U 2 n and 2 k1 , all functions in the form Vðx, tÞ ¼ uð x li ðU,ÞtÞRi ðU, Þ,
(3)
depicting waves with amplitude collinear to Ri(U, ), travelling in the direction with speed li(U, ), are solutions of the system resulting from linearizing (1) about U: k X (4) @t V + DFa ðUÞ@a V ¼ 0: a¼1
The simplest example is provided by the scalar conservation law k X @t u + @a fa ðuÞ ¼ 0:
(5)
a¼1
However, the primordial, and still most important, example is the system of the Euler equations @t r + divðrvÞ ¼ 0 (6) @t ðrvÞ + divðrv vÞ + grad pðrÞ ¼ 0, which govern the isentropic flow of a gas. In (6), r denotes the (mass) density, v stands for velocity and p is the pressure. The gradient and divergence operate with respect to the spatial variable. Eq. (6)1 expresses conservation of mass while (6)2 states conservation of (linear) momentum. The system is hyperbolic so long as p0 (r) > 0. The notion of entropy plays a very important role in the theory of hyperbolic conservation laws. A scalar function (U) is called an entropy for the system (1), associated with the entropy flux qa(U), a ¼ 1,…, k, if for U 2 n Dqa ðUÞ ¼ DðUÞDFa ðUÞ, a ¼ 1,…, k:
(7)
This is equivalent to requiring that any smooth solution U of (1) satisfies automatically the additional conservation law k X (8) @t ðUÞ + @a qa ðUÞ ¼ 0: a¼1
In the scalar case (5), anyR function (u) qualifies as entropy, with associated entropy flux qa ðuÞ ¼ 0 ðuÞf 0 ðuÞdu, a ¼ 1,…,k. Also, for k ¼ 1 and n ¼ 2, the system (7) of two equations in two unknowns, yields a rich family
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of entropy–entropy flux pairs (, q). In all other cases, however, since kn > k +1, (7) is overdetermined so one should not expect the existence of nontrivial entropies for generic systems (1). Nevertheless, it turns out that in virtually all interesting systems arising in physics, the fluxes are judiciously selected so that an entropy exists. Moreover, quite often, though not always, this entropy is a convex function of U. A case in point is the system (6) of the Euler equations, which is equipped with the entropy–entropy flux pair 1 1 (9) ¼ reðrÞ + rjvj2 , q ¼ ½reðrÞ + rjvj2 + pðrÞv, 2 2 R 2 where eðrÞ ¼ r pðrÞdr is the internal energy. It turns out that the hyperbolicity condition p0 (r) > 0 renders convex, as a function of the canonical state vector (r, rv). The extra conservation law (8) here expresses conservation of mechanical energy. Any symmetric system (1) is endowed with the entropy–entropy flux pair 1 (10) ðUÞ ¼ jUj2 , qa ¼ U Fa ðUÞ ’a ðUÞ, 2 where ’a is a ‘potential’ with D’a ¼ Fa, which exists since DFa is symmetric. It is easily seen that, conversely, any system of conservation laws (1) possessing a convex entropy (U) is symmetrized by introducing the new state vector V ¼ D(U). In particular, any such system is hyperbolic. The Cauchy problem is locally well posed for any system of conservation laws (1) endowed with a convex entropy. Indeed, for any function U0 in the 1 Sobolev space H l ðk Þ, with l > k + 1, there exists a unique C1 solution U 2 of (1) defined on k ½0, T∞ Þ and satisfying the initial condition Uðx, 0Þ ¼ U0 ðxÞ, x 2 k :
(11)
The lifespan T∞ is maximal in the sense that either T∞ ¼ ∞ or else T∞ < ∞, in which case maxk jrUð ,tÞj ! ∞, as t ! T∞ . The proof of the above proposition rests on establishing a priori bounds on the L2 ðk Þ norms of U and all of its spatial derivatives up to order l. These are derived by means of ‘energy’ estimates induced by the presence of the convex entropy. For instance, (8) yields a bound on the L2 ðk Þ norm of U. The 1 restriction l > k + 1 is needed in order to keep k rUkL∞ bounded, in which 2 case the family of energy estimates closes. It turns out that for nonlinear hyperbolic systems of conservation laws the case of finite lifespan for smooth solutions is the rule rather than the exception. This comes as a result of the wave breaking effect: As waves move at different speeds, “compressive” wave profiles get progressively steeper and eventually break. This scenario is easily seen in the setting of the scalar conservation law. Assume u(x, t) is a local smooth solution of the Cauchy problem for (5), with initial values u(x, 0) ¼ u0(x). Consider characteristics
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associated with u as trajectories of the ordinary differential equation dxa ¼ f 0 ðuðx,tÞÞ, a ¼ 1, …, k. Letting an overdot denote the derivative dt Pa fa0 ðuÞ@a in the characteristic direction, we may write (5) as u_ ¼ 0, @t + which shows that characteristics are straight lines along which u is constant. Thus, with any (x, t) is associated y 2 k such that ya ¼ xa tfa0 ðuðx,tÞÞ, a ¼ 1, …,k
(12)
and u(x, t) ¼ u0(y). This easily implies @b uðx,tÞ ¼ P
@b u0 ðyÞ , k X 0 1 + t @a fa ðu0 ðyÞÞ a¼1
b ¼ 1, …,k:
(13)
@a fa0 ðu0 ðyÞÞ takes negative values at any points, 1 attaining a negative minimum, say e, on k , ru must blow up at t ¼ . e Wave breaking is particularly pronounced in one space dimension, where waves are confined and cannot avoid interacting with one another. In several space dimensions, depending on the geometry of the system, wave breaking may be impeded by wave dispersion, which has the opposite effect. As we saw earlier, this is not the case for the scalar conservation law (5). The situation is quite different for the Euler equations (6), in three spatial dimensions. For this system, wave breaking and dispersion are evenly matched and their competition is very keen. Dispersion manages to prolong thelifespan of 1 smooth solutions with initial derivatives of size e to O exp —much longer e 1 lifespan of smooth solutions to the scalar conservation law. than the O e Nevertheless, eventually wave breaking prevails and the derivatives of the solution blow up. In view of the breakdown of smooth solutions, in order to get solutions in the large, one has to resort to weak, distributional solutions, namely bounded measurable vector fields U on some domain X of k , which satisfy Z Z k X (14) ½@t fU + @a fFa ðUÞdxdt ¼ 0, We conclude that if
X
a¼1
for all smooth test functions f with compact support in X . Notice that it is possible to define weak solutions because conservation laws (1) are in divergence form. In particular, let us seek weak solutions defined on all of k in the form 8 < U for x st < 0 Uðx, tÞ ¼ (15) : U + for x st > 0,
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where U, U+ are constant states in n , 2 k1 and s is a scalar. It is a simple exercise to verify that U in (15) will satisfy (14) for all test functions f if and only if k X (16) a ½Fa ðU + Þ Fa ðU Þ s½U + U ¼ 0: a¼1
Recalling the definition of hyperbolicity one sees that for any fixed U 2 n and 2 k1 it is possible to find U+ in the vicinity of U such that U+ U is nearly collinear to Ri(U, ) and (16) holds for some s near li(U, ). Such a weak solution of (1) is termed a planar shock wave with amplitude U+ U propagating in the direction with speed s, and (16) is called the Rankine–Hugoniot jump condition. More generally, there exist piecewise smooth weak solutions to (1) with jump discontinuities across curved shocks. In that situation (16) still holds across the shock, though now , U, U+ and s are no longer constant, as they may vary from point to point on the shock. Familiarity with weak solutions to hyperbolic conservation laws is enhanced by visualizing them as composites of continuous waves and shock waves, with the understanding that these two constituents may be finely blended. Though it is not presently known whether every L∞ weak solution fits the above description, this is certainly the case at least for solutions U of class BV, for which @ tU and @ aU are Radon measures. Indeed, the domain of U 2 BV is the union of three disjoint sets, namely: (a) the set of points of approximate continuity, in the sense of Lebesgue; (b) the set of points of (approximate) jump discontinuity, which is an at most countable family of disjoint C1k-dimensional manifolds, across which the jump condition (16) holds; and (c) a ‘small’ residual set whose k-dimensional Hausdorff measure is nil. As we shall see in the next section, a serious obstacle for dealing with weak solutions is the loss of uniqueness in the Cauchy problem. Accordingly, additional requirements must be imposed, in order to weed out spurious solutions and single out the unique admissible one. In what follows, we outline two methods in that direction, which are dictated, or at least motivated, by physics. Assume our system (1) is endowed with an entropy–entropy flux pair (, q), with D2(U) positive definite. Recall that under such conditions any smooth solution of (1) satisfies automatically the extra conservation law (8). However, this is no longer the case for weak solutions of (1). We now stipulate that a bounded measurable weak solution U of (1) satisfies the entropy admissibility condition, relative to , on a domain X of k , if k X (17) @t ðUÞ + @a qa ðUÞ 0 a¼1
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holds in the sense of distributions, that is Z Z k X ½@t cðUÞ + @a cqa ðUÞdxdt 0, X
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(18)
a¼1
for all nonnegative smooth test functions c, with compact support in X . In particular, all smooth solutions of (1) satisfy this condition and thus are deemed admissible. In the physical applications, the inequality (17) typically manifests, directly or indirectly, the second law of thermodynamics. The entropy admissibility condition is particularly effective for scalar conservation laws (5), for which, as we saw earlier, any convex function (u) may serve as entropy. The approach pioneered by Kruzkov is to deem a weak solution u of (5) admissible if it satisfies k X (19) @t ðuÞ + @a qa ðuÞ 0 a¼1
R for all convex functions (u) and qa ðuÞ ¼ 0 ðuÞf 0 ðuÞdu. It has been shown that for any u0 ðxÞ 2 L∞ ðk Þ there exists a unique admissible solution u(x, t) of (5) on k ½0, ∞Þ, with initial value u0(x). Furthermore, admissible solutions are strongly stable as they have the following L1 contraction property: Z Z juðx,tÞ uðx,tÞjdx ju0 ðxÞ u0 ðxÞjdx, 0 t < ∞, (20) k
k
holds for any pair ðu, uÞ of admissible solutions with initial values ðu0 , u0 Þ: In particular, applying (20) with u0 ðxÞ ¼ u0 ðx + eÞ and thereby uðx, tÞ ¼ uðx + e,tÞ, where e is an arbitrary k-vector, we deduce that initial data u0 of class BV generate solutions u to the Cauchy problem for the scalar conservation law (5) that are also of class BV and the variation of u(, t) over k is a nonincreasing function of t on ½0, ∞Þ. By contrast, systems of conservation laws (1) with n 2 typically possess a single convex entropy so that the admissibility condition (17) does not generally suffice for uniqueness of solutions to the Cauchy problem. In particular, it has been shown that for certain initial data the Cauchy problem for the system of the Euler equations (6) admits infinitely many weak solutions satisfying the entropy admissibility condition (17), relative to the entropy (9). Nevertheless, it turns out that even a single inequality (17) for a convex entropy suffices for securing uniqueness and stability of smooth solutions to the Cauchy problem, not only within the class of smooth solutions but even within the broader class of admissible weak solutions (so-called weak-strong tÞ, stability). Specifically, if on k ½0, T there exist a smooth solution Uðx, with initial values U0 ðxÞ, and also a weak solution U(x, t), with initial values U0(x), satisfying the entropy admissibility condition (17), then Z Z 2 jUðx, tÞ Uðx,tÞj dx ceat jU0 ðxÞ U0 ðxÞj2 dx, 0 t T: (21) k
k
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The proof of this property is established with the help of the so-called relative entropy ¼ ðUÞ ðUÞ DðUÞ½U HðU, UÞ U:
(22)
It is easy to see that for solutions of the form (15) the entropy admissibility condition (18) reduces to the jump condition k X
a ½qa ðU + Þ qa ðU Þ s½ðU + Þ ðU Þ 0:
(23)
a¼1
For arbitrary weak solutions U satisfying the entropy admissibility condition, the left-hand side of (17) is a nonpositive distribution, and thereby a measure. In particular, it turns out that when U is of class BV the above measure is concentrated on the set of points of jump discontinuity and (17) reduces to the requirement that (23) holds across every shock. This has motivated the widely held conjecture that the admissibility of weak solutions hinges exclusively on a localized test, such as (23), to be applied to every point of jump discontinuity, involving just U, U+ and s, interrelated through (16). Even though it may not be universally valid, the above premise enjoys wide applicability, as we shall see in the next section. We now turn to an alternative admissibility criterion for weak solutions, which is also motivated by physics. The isentropic flow of a viscous gas is governed by the Navier–Stokes equations 8 < @t r + divðrvÞ ¼ 0 (24) : @t ðrvÞ + divðrv vÞ + grad pðrÞ ¼ ðl + mÞgrad div v + mDv, where l and m are viscosity coefficients. Any gas in nature has some, perhaps minute, viscosity and hence admissible solutions to the Euler equations (6) should be viewed as asymptotic solutions to the Navier–Stokes equations (24), with viscosity coefficients tending to zero. One may extend the above argument to general systems (1) as follows. Next to (1), we consider the system @t U +
k X
@a Fa ðUÞ ¼ mDU,
(25)
a¼1
where m is a positive ‘viscosity’ parameter, and postulate that a weak solution U of (1) satisfies the viscosity admissibility condition if it is the m ! 0 limit of smooth solutions Um of the parabolic system (25). Assume (1) is endowed with an entropy–entropy flux pair (, q), with (U) convex. Suppose {Um} is a family of solutions of (25), with Um converging, boundedly almost everywhere, as m ! 0, to a weak solution U of (1). Multiplying (25) by D(Um) and using (7) yields
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@t ðUm Þ +
k X
k X
@a qa ðUm Þ ¼ mDðUm Þ m
a¼1
a¼1
@a Um> D2 ðUm Þ@a Um :
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(26)
As m ! 0, the left-hand side of (26) converges, in the sense of distributions, to the left-hand side of (17). On the right-hand side of (26), the first term converges to zero while the second term stays nonpositive, since (U) is convex. We thus conclude that U satisfies (17), i.e. the viscosity admissibility condition implies the entropy admissibility condition. However, as we shall see in the next section, the converse is not generally true. In the scalar case, for consistency with the notation in (5), we write (25) in the form k X (27) @t u + @a fa ðuÞ ¼ mDu: a¼1
Solutions to the Cauchy problem for (27) have the L1 contraction property (20). This estimate serves as the tool for showing that, as m ! 0, the solution um(x, t) of the Cauchy problem for (27), with initial data u0(x) in L∞ , converges, boundedly almost everywhere, to the unique weak solution u(x, t) of (5), with the same initial values, which satisfies (19) for all convex functions (u). Thus for scalar conservation laws, the viscosity admissibility condition is equivalent to the entropy admissibility condition, for all convex entropies. By contrast, for n 2, the convergence of solutions of (25), as m ! 0, has been established only under quite restrictive hypotheses. Consequently, there is no straightforward way to test whether any particular solution of (1) satisfies the viscosity admissibility condition. In practice, the testing is performed for solutions of the form (15), with the expectation that local admissibility of shocks renders the entire solution admissible. Accordingly, one seeks to capture the solution (15) as the m ! 0 limit of solutions Um of (25) in the form x st , Um ðx,tÞ ¼ VðxÞ, x ¼ (28) m depicting fronts propagating in the direction with speed s, and becoming progressively steeper, as m decreases. Then V must satisfy the ordinary differential equation k X (29) V€ ¼ sV_ + a F_a ðVÞ a¼1
on ð∞, ∞Þ, with boundary conditions Vð∞Þ ¼ U ,Vð∞Þ ¼ U + . Integrating (29), k X (30) a ½Fa ðVÞ Fa ðU Þ: V_ ¼ s½V U + a¼1
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Notice that the right-hand side of (30) vanishes both at V ¼ U and V ¼ U+, the latter by virtue of the jump condition (16). We conclude that the solution (15) of (1) satisfies the viscosity admissibility condition if there exists an orbit of (30) joining the equilibrium points U and U+. The function V (x) is called the shock profile or the shock structure. It allows us, so to say, to observe the shock under the microscope. There are many other topics of interest in the theory of hyperbolic systems of conservation laws. When dealing with solutions of (1) on some domain of k the question of assigning proper boundary conditions requires deep analysis, even in the context of smooth solutions. One often encounters in the applications so-called systems of balance laws k X (31) @t U + @a Fa ðUÞ ¼ GðUÞ, a¼1
with a source term G(U) manifesting relaxation. In that case, the source has a dissipative effect inducing global existence of smooth solutions to the Cauchy problem, when the initial data are smooth and ‘small’. As noted earlier, the Cauchy problem for scalar conservation laws, n ¼ 1, is well-posed, in the setting of admissible weak solutions, for any k 1. On the other hand, virtually nothing is known on the existence and uniqueness of weak solutions to the Cauchy problem for systems, n 2, when k 2. However, considerable progress has been made for systems in one spatial dimension, k ¼ 1. The following section will provide an overview.
3 STRICTLY HYPERBOLIC SYSTEMS IN ONE SPATIAL DIMENSION This section surveys aspects of the theory of hyperbolic systems of conservation laws @t U + @x FðUÞ ¼ 0 (32) in one spatial dimension. As in Section 2 , the state vector U and the flux F(U) take values in n . The system is called strictly hyperbolic if for any U 2 n the Jacobian matrix DF(U) possesses n real distinct eigenvalues l1(U) < ⋯ < ln(U), called characteristic speeds, and thereby linearly independent eigenvectors R1 ðUÞ,…, Rn ðUÞ: The theory of hyperbolic systems that are not strictly hyperbolic is still incomplete, even for the simplest case where two characteristic speeds coalesce in just a single point of n . In our discussion we will employ as demonstration models the scalar conservation law @t u + @x f ðuÞ ¼ 0 and the so-called p-system
(33)
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8 < @t u @x v ¼ 0 :
(34) @t v + @x pðuÞ ¼ 0,
which p is ffiffiffiffiffiffiffiffiffiffiffiffiffi strictly whenffi p0 (u) < 0, with characteristic speeds ffi hyperbolic pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 l1 ¼ p ðuÞ and l2 ¼ p0 ðuÞ. The p-system is the Lagrangian version of the Euler equations (6), in one spatial dimension. It governs the rectilinear flow of a gas in a duct, with u standing for specific volume (the inverse of density) and v denoting velocity. The same system governs the one-dimensional motion of elastic solids (longitudinal vibrations of a bar, shearing motion of a slab, oscillation of a string, etc.). In that context one usually replaces in (34) the pressure p with the negative stress s. The contrast in behaviour between linear and nonlinear hyperbolic systems (32), in one spatial dimension, is particularly pronounced when the latter satisfy the genuine nonlinearity condition, namely, after normalizing the eigenvectors Ri, (35) Dli ðUÞRi ðUÞ ¼ 1, i ¼ 1, …, n, U 2 n : In particular, the scalar conservation law (33) is genuinely nonlinear when f 00 (u) 6¼ 0, for all u. The simplest example is the celebrated Burgers equation 1 @t u + @x u2 ¼ 0, (36) 2 which is ubiquitous in the area of conservation laws. Despite its apparent simplicity, (36) exemplifies many of the principal features of genuinely nonlinear systems of hyperbolic conservation laws and provides an excellent model for an initial approach to the subject. The p-system (34) is genuinely nonlinear when p00 (u) 6¼ 0, for all u. This is the natural assumption for fluids, where p00 (u) > 0, while in solids the stress function s(u) may have inflection points. As we saw in the previous section, smooth solutions to the Cauchy problem for the scalar conservation law, in any number of spatial dimensions, typically break down in finite time because of wave breaking. This process is particularly transparent in the setting of the Burgers equation. Assume u is a smooth solution to the Cauchy problem for (36), defined on dx ð∞, ∞Þ ½0, TÞ. Consider the characteristics ¼ uðx, tÞ and denote differdt entiation @ t + u@ x in the characteristic direction by an overdot. Differentiation of (36) with respect to x yields that the derivative v ¼ @ xu satisfies the Bernoulli equation v_ + v2 ¼ 0, along the characteristic. Thus if v(0) ¼ e < 0, 1 @ xu must blow up at time t ¼ . The same mechanism is responsible for the e blowing up of smooth solutions to the Cauchy problem for general genuinely nonlinear hyperbolic systems of conservation laws in one spatial dimension,
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but the proof is considerably more complicated. One should expect generic blowing up of smooth solutions even for nonlinear hyperbolic systems of conservation laws that are not genuinely nonlinear, but this needs to be demonstrated individually for each case. In view of the above, to solve the Cauchy problem in the large one has to resort to weak solutions that contain shocks. The Rankine–Hugoniot jump condition (16), satisfied across shocks, here reduces to FðU + Þ FðU Þ s½U + U ¼ 0:
(37)
The standard terminology is that ‘U+ is joined to U by a shock of speed s’. In particular, for the scalar conservation law (33), s¼
f ðu + Þ f ðu Þ : u + u
(38)
We now demonstrate, in the context of the Burgers equation, the loss of uniqueness of weak solutions, reported in Section 2. Indeed, the Cauchy problem for (36) with initial data u(x, 0) ¼ 1, for x < 0, and u(x, 0) ¼ 1, for x > 0, admits infinitely many weak solutions, including the following two: 8 x > 1 ∞ < < 1 > > t < x x 1 1 (39) uðx, tÞ ¼ > t t > x > : 1 1 < < ∞, t 8 x < 1 ∞ < < 0 t uðx,tÞ ¼ (40) x : 1 0 < < ∞: t This pathology is encountered in every nonlinear system (32) of conservation laws. In order to weed out spurious solutions, so as to restore uniqueness to the Cauchy problem, we appeal to the admissibility criteria introduced in Section 2, as related to shocks. We begin with the case of the scalar conservation law. Consider a shock for (33) joining the states u and u+. Its speed s is given by (38). Eq. (30) for the structure of the shock here takes the form v_ ¼ sðv u Þ + f ðvÞ f ðu Þ:
(41)
The shock will satisfy the viscosity admissibility condition if there exists a solution v(t) of (41) on ð∞, ∞Þ, with vð∞Þ ¼ u and vð∞Þ ¼ u + . Thus the right-hand side of (41) should not change sign between u and u+, or equivalently 8 < 0 if u < v < u + f ðvÞ hðvÞ (42) : 0 if u + < v < u ,
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where h(v) is the straight line segment (chord), with slope s, connecting the points (u, f(u)) and (u+, f(u+)) on the graph of f. This is the celebrated Oleinik E-condition. Looking at the same question from the perspective of the entropy admissibility condition, we fix any convex R function (u) as entropy and determine the associated entropy flux qðuÞ ¼ 0 ðuÞf 0 ðuÞdu: After a simple calculation, (23) here yields Z u+ 00 ðvÞ½ f ðvÞ hðvÞdv ¼ qðu + Þ qðu Þ s½ðu + Þ ðu Þ 0: (43) u
It is clear that the Oleinik E-condition (42) is equivalent to (43), for all convex (u). On the other hand, (43) for just one convex (u) does not necessarily imply (42), unless f 00 (u) 6¼ 0, for all u, i.e. when (33) is genuinely nonlinear. In the genuinely nonlinear case, Eqs. (42) and (43) reduce to the celebrated Lax E-condition f 0 ðu + Þ < s < f 0 ðu Þ:
(44)
In particular, for the Burgers equation (36), the Lax E-condition (44) reduces to u+ < u, so that the solution (40) violates the viscosity and the entropy admissibility conditions. In fact (39) is the unique admissible solution to that Cauchy problem. We now turn to the question of admissibility of shocks for general strictly hyperbolic systems of conservation laws (32). The first task is to determine the Hugoniot locus, associated with a fixed state U, which consists of all states U+ that satisfy (37) for some s, and thereby may be joined to U by a shock. The part of the Hugoniot locus contained in a small neighbourhood of U has a definite and simple structure. Notice that when (37) holds with jU+ Uj small, then s must be close to one of the characteristic speeds li(U) and U+ U must be almost collinear to the associated eigenvector Ri(U). Indeed, since (32) is assumed to be strictly hyperbolic, it follows by standard bifurcation theory that for each i ¼ 1,…, n there exists a smooth curve Wi(t) in some neighbourhood of U and associated scalar function si(t) such that FðWi ðtÞÞ FðU Þ si ðtÞ½Wi ðtÞ U ¼ 0,
(45)
Wi ð0Þ ¼ U ,
(46)
W_ i ð0Þ ¼ Ri ðU Þ,
(47)
si ð0Þ ¼ li ðU Þ,
(48)
1 s_i ð0Þ ¼ Dli ðU ÞRi ðU Þ: 2
(49)
Wi(t) is called the i-shock curve of (1) through U and the Hugoniot locus near U is the union of the n shock curves.
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As an example, consider the p-system (34), with Rankine–Hugoniot conditions (37) in the form 8 < v + v + sðu + u Þ ¼ 0 (50) : pðu + Þ pðu Þ sðv + v Þ ¼ 0, whence
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðu + Þ pðu Þ : s¼ u + u
(51)
It is here convenient to parametrize the shock curve through (u, v) by u, in which case v ¼ v sðu u Þ, (52) taking s from (51), with the negative square root for the 1-shock curve and the positive square root for the 2-shock curve. For the above simple system, (52) describes not only the local but even the global Hugoniot locus. In general, however, the global portrait of the Hugoniot locus may be quite complex, containing detached branches, islands, etc. When the system is not strictly hyperbolic, even the local portrait of the Hugoniot locus may be geometrically varied and complex. Accordingly, here we shall limit discussion to systems (32) that are strictly hyperbolic and for shocks of small amplitude. The i-shocks must pass the admissibility test. For the viscosity admissibility condition, (30) here reduces to V_ ¼ s½V U + FðVÞ FðU Þ:
(53)
One seeks a solution V(t) of the ordinary differential equation (53) on ð∞, ∞Þ, with boundary conditions Vð∞Þ ¼ U , Vð∞Þ ¼ U + . On the other hand, if (32) is endowed with an entropy–entropy flux pair (, q), with (U) convex, the entropy admissibility condition (23) for the shock here reduces to qðU + Þ qðU Þ s½ðU + Þ ðU Þ 0:
(54)
As in the scalar case, it turns out that the viscosity condition always implies the entropy condition. Furthermore, whenever the system (32) is genuinely nonlinear (35), then an i-shock of small amplitude satisfies the viscosity condition if and only if it satisfies the entropy condition, and also if and only if the Lax E-condition li ðU + Þ < s < li ðU Þ (55) holds. Thus, with reference to the i-shock curve and by virtue of (48) and (49), we deduce that, for genuinely nonlinear systems, an i-shock with small amplitude will be admissible when U+ ¼ Wi(t), for t < 0, and inadmissible when U+ ¼ Wi(t), for t > 0.
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When the system (32) is not genuinely nonlinear, the viscosity admissibility condition still manages to weed out all inadmissible shocks, whereas the entropy condition and the Lax E-condition are less selective, being merely necessary but not sufficient for admissibility. A strengthening of the Lax E-condition that is equivalent to the viscosity condition, even when the system is not genuinely nonlinear, is provided by the Liu E-condition, which stipulates that an i-shock joining U+ ¼ Wi(t) to U will be admissible if si(s) si(t), for all s between 0 and t. Thus, the Liu E-condition generalizes the Oleinik E-condition, from scalar conservation laws to strictly hyperbolic systems of conservation laws. The issue of admissibility of shocks in systems that are not strictly hyperbolic is still unresolved. An important feature of systems (32) is that they are invariant under uniform stretching of the space-time variables, as a result of which they admit self-similar solutions in the form x Uðx,tÞ ¼ VðxÞ, x ¼ : (56) t is a trivial case of a self-similar solution. Any constant state, VðxÞ ¼ U, Another important example is provided by step functions, V(x) ¼ U, for x < s, and V(x) ¼ U+, for x > s, where U and U+ satisfy (37) and thus may be joined by a shock of speed s. Next we consider the possibility of Lipschitz continuous self-similar solutions (56), with V(x) ¼ U, for x x, V(x) ¼ U+, for x x+, and V(x) smooth, for x x x+. In that case, one says that ‘U+ is joined to U by a centred rarefaction wave’—terminology borrowed from gas dynamics. We investigate this question under the assumption that the system (32) is genuinely nonlinear (35). Substituting from (56) into (32) yields ½DFðVðxÞÞ xIV_ ðxÞ ¼ 0:
(57)
li ðVðxÞÞ ¼ x,
(58)
Thus, assuming V_ ðxÞ 6¼ 0, and V_ ðxÞ is collinear to Ri(V (x)), for some i ¼ 1,…, n (i-rarefaction wave). Differentiating (58) with respect to x and using (35) yields V_ ¼ Ri ðVÞ:
(59)
Thus x ¼ li(U), x+ ¼ li(U+) and U+ ¼ V(x+), where V(x) is the solution of the ordinary differential equation (59) with initial value V(x) ¼ U. We reparametrize V(x), replacing x with the new parameter t ¼ xx, denote it by Vi(t) and call it the i-rarefaction curve through U. As an example, consider the p-system (34), under the assumption p00 (u) > 0 of genuine nonlinearity. As with the shock curve (52), it is convenient to parametrize the rarefaction curve through (u, v) by u, in which case it assumes the form
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Z u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ðwÞdw, v ¼ v
(60)
u
where the minus sign gives the 1-rarefaction curve and the plus sign yields the 2-rarefaction curve. Fig. 1 provides a sketch of both the i-shock curve and the i-rarefaction curve through U. Notice that if we discard the part of the shock curve associated with inadmissible shocks and concatenate the admissible part of the i-shock curve with the i-rarefaction curve, we end up with a C1 curve called the i-wave curve through U. The celebrated Riemann Problem is the Cauchy problem for (32) with initial data ∞ < x < 0 Ul (61) Uðx, 0Þ ¼ Ur 0 < x < ∞, where Ul and Ur are given constant states. Since the above initial data are invariant under stretching of the x-variable, one expects that the solution to the Riemann Problem will be self-similar, (56). As (32) remains invariant under extreme stretches and extreme contractions of the space-time variables, it is expected that the solution of the Riemann problem should depict both the local and the large time behaviour of admissible weak solutions to the Cauchy problem, under general initial conditions. One might say that the solution of the Riemann problem provides the instrument for observing general solutions ‘under the microscope’ as well as ‘through the telescope’. Furthermore, the solution to the Riemann problem has been used as a building block for constructing general solutions, for theoretical or numerical purposes. For genuinely nonlinear systems, it is possible to synthesize the solution to the Riemann problem with the help of the wave curves, introduced above. Fig. 2 shows how this is done for the case of the p-system (34). Given the endstates (ul, vl) and (ur, vr), one has to locate an ‘intermediate’ state (um, vm) with the property that (um, vm) lies on the 1-wave curve through (ul, vl) and, in turn, (ur, vr) lies on the 2-wave curve through (um, vm). This is possible because 1- and 2-wave curves intersect transversely, by strict hyperbolicity. The resulting solution to the Riemann Problem consists of the three constant states (ul, vl), (um, vm), (ur, vr) together with a 1-wave joining (um, vm) to
FIG. 1 Shock and rarefaction wave curves.
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A
B
FIG. 2 Solution to the Riemann problem.
(ul, vl) and a 2-wave joining (ur, vr) to (um, vm). Because of the relative location of (ul, vl), (ur, vr) in Fig. 2 , the 1-wave turns out to be a shock and the 2-wave a rarefaction. However, any combination is possible. The situation is similar for general genuinely nonlinear systems (32). By means of the implicit function theorem, one shows that for any given states Ul and Ur, with jUr Ulj sufficiently small, there exists a unique solution to the Riemann problem, with initial data (61), which is composed of n + 1 constant states Ul ¼ U0, U1,…, Un ¼ Ur, with Ui joined to Ui1 by either an admissible i-shock or by an i-rarefaction wave. The solution to the Riemann problem for strictly hyperbolic systems (32) that are not genuinely nonlinear has a similar structure, involving as above n +1 constant states Ul ¼ U0, U1,…, Un ¼ Ur. However, now Ui is joined to Ui1 by an i-wave that is no longer necessarily a single i-shock or i-rarefaction wave, but possibly a composite of several such shocks and rarefactions. For systems that are not strictly hyperbolic, the solution to the Riemann problem may assume a variety of forms and the issue of admissibility and uniqueness is not completely resolved. We now turn to the general Cauchy problem for strictly hyperbolic systems (32), under initial conditions Uðx,0Þ ¼ U0 ðxÞ, ∞ < x < ∞:
(62)
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The state of the art may be summarized as follows: If the total variation of U0(x) is sufficiently small, there exists a unique admissible weak solution U(x, t) of (32), (62) on the upper half-plane. The function U is in BV with respect to the space-time variables. Furthermore, for each fixed t, the function U(, t) has bounded variation on ð∞, ∞Þ and TVð∞, ∞Þ Uð , tÞ cTVð∞, ∞Þ U0 ð Þ,
0 t < ∞:
(63)
The above solution has been constructed by three methods, namely: (a) The random choice method of Glimm, with essential extensions by Tai-Ping Liu. It uses solutions to the Riemann problem as building blocks for approximating the general solution. (b) The front tracking algorithm developed by the Italian School headed by Bressan. The strategy in this approach is to construct approximate solutions that contain only constant states and shocks, by replacing rarefaction waves with fans of inadmissible, albeit very weak, rarefaction shocks. Shock interactions are then handled by solving Riemann problems. (c) The vanishing viscosity approach of Bianchini and Bressan. In all three methods, the crucial step is to establish the bound (63) on the approximate solutions, for some uniform constant c. For the above theorem to hold, the restriction that the total variation of the initial data must be small is essential. Indeed, there are cases of systems in which weak solutions under initial data with large total variation break down in finite time. A major challenge to the theory at the present time is to determine whether the systems of hyperbolic conservation laws arising in physics are free from such pathologies. An even greater challenge is the issue of existence of weak solutions in the large, for systems in more than one spatial dimension: This is currently terra incognita.
REFERENCES Bressan, A., 2000. Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford University Press, Oxford. Dafermos, C.M., 2016. Hyperbolic Conservation Laws in Continuum Physics, fourth ed. Springer, Heidelberg. Holden, H., Risebro, N.H., 2015. Front Tracking for Hyperbolic Conservation Laws, second ed. Springer, New York. Serre, D., 1999. Systems of Conservation Laws, vols. 1–2. Cambridge University Press, Cambridge. Smoller, J.A., 1994. Shock Waves and Reaction-Diffusion Equations, second ed. Springer, New York.
Introduction to the Theory of Hyperbolic Conservation Laws C.M. Dafermos Division of Applied Mathematics, Brown University, Providence, RI, United States
Chapter Outline 1 Introduction 2 Basic Structure of Hyperbolic Conservation Laws
1 2
3 Strictly Hyperbolic Systems in One Spatial Dimension References
10 18
ABSTRACT This is a brief, informal introduction to nonlinear hyperbolic conservation laws, underscoring their inherent properties (wave breaking, entropy conditions) and sketching the state of the art in their analysis. Keywords: Hyperbolic conservation laws, Entropy, Viscosity, Shocks, Riemann Problem AMS Classification Codes: 35L65, 35L67
1
INTRODUCTION
The conservation laws of gas dynamics, elastodynamics, electrodynamics and other branches of classical physics are typically expressed by hyperbolic partial differential equations or systems thereof. In particular, it is hyperbolic systems that provide the proper mathematical setting for a host of wave phenomena. The salient feature of solutions to nonlinear hyperbolic systems resulting from conservation laws is wave breaking, which triggers the development of jump discontinuities that propagate on as shock waves. This renders the mathematical theory particularly hard, as it must cope with weak solutions. The difficulty is exacerbated by the fact that uniqueness and stability are lost in the realm of weak solutions. As a remedy, one seeks selection criteria, motivated by physical or mathematical considerations, that hopefully weed out all spurious solutions, singling out the admissible one. In consequence of these difficulties, and despite considerable progress achieved over the past 50 years, Handbook of Numerical Analysis. http://dx.doi.org/10.1016/bs.hna.2016.08.003 © 2016 Elsevier B.V. All rights reserved.
1
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central mathematical issues are still wide open, especially in several spatial dimensions. It is fair to admit that at the present time computation has outpaced the theory. The aim of this introductory chapter is to provide a description of the basic structure of hyperbolic systems of conservation laws and to survey their inherent properties. We shall not provide specific bibliographic references. The reader who seeks more detailed information should consult one or more of numerous existing texts. The list includes Smoller (1994) (a clear introduction to the basics, albeit somewhat dated), Holden and Risebro (2015) (a readable, successful marriage of theory with numerics), Serre (1999) (an insightful introduction to the basics, supplemented with an interesting selection of more advanced topics), Bressan (2000) (a nice exposition starting out at an introductory elementary level and becoming progressively more focused and technical) and Dafermos (2016) (an encyclopaedic coverage of the field, with voluminous bibliography). The perspective and style of presentation in this chapter are borrowed from Dafermos (2016). The author is indebted to Zheng Sun for his valuable assistance in drawing the figures.
2 BASIC STRUCTURE OF HYPERBOLIC CONSERVATION LAWS The canonical form of a system of n conservation laws in k spatial dimensions reads k X (1) @t U + @a Fa ðUÞ ¼ 0: a¼1
The (unknown) n-dimensional state vector field U is a function of the k-dimensional spatial variable x and the scalar temporal variable t. For a ¼ 1,…, k, the flux Fa(U) is a given smooth function from n to n and @ a stands for @/@xa. The terminology, with origins in classical physics, stems from the observation that (1) holds on some domain of k if and only if Z I X k d (2) Udx + a Fa ðUÞdS ¼ 0 dt O @O a¼1 for all smooth subdomains O, with @O denoting the boundary of O and standing for the exterior unit normal on @O. Indeed, the ‘physical’ interpretation of (2) is that the n-vector valued quantity with density U is conserved, in the sense that the rate of change in the amount stored in O is balanced by the rate of flux, in or out of O, through @O. In what follows we will employ matrix notation, identifying n with column vectors n1 . The symbol D will denote the gradient operator in n , mapping scalar fields into 1n row vector fields and n1 column vector fields into nn matrix fields.
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The system of conservation laws (1) will Pbe called hyperbolic if for any fixed U 2 n and 2 k1 , the n n matrix ka¼1 a DFa ðUÞ has real eigenvalues l1(U, ),…, ln(U, ) and an associated set of n linearly independent eigenvectors R1(U, ),…, Rn(U, ). An important subclass of hyperbolic conservation laws are the symmetric systems (1) with the property that for a ¼ 1,…, k and any U 2 n , the matrices DFa(U) are symmetric. As a consequence of hyperbolicity, for any fixed U 2 n and 2 k1 , all functions in the form Vðx, tÞ ¼ uð x li ðU,ÞtÞRi ðU, Þ,
(3)
depicting waves with amplitude collinear to Ri(U, ), travelling in the direction with speed li(U, ), are solutions of the system resulting from linearizing (1) about U: k X (4) @t V + DFa ðUÞ@a V ¼ 0: a¼1
The simplest example is provided by the scalar conservation law k X @t u + @a fa ðuÞ ¼ 0:
(5)
a¼1
However, the primordial, and still most important, example is the system of the Euler equations @t r + divðrvÞ ¼ 0 (6) @t ðrvÞ + divðrv vÞ + grad pðrÞ ¼ 0, which govern the isentropic flow of a gas. In (6), r denotes the (mass) density, v stands for velocity and p is the pressure. The gradient and divergence operate with respect to the spatial variable. Eq. (6)1 expresses conservation of mass while (6)2 states conservation of (linear) momentum. The system is hyperbolic so long as p0 (r) > 0. The notion of entropy plays a very important role in the theory of hyperbolic conservation laws. A scalar function (U) is called an entropy for the system (1), associated with the entropy flux qa(U), a ¼ 1,…, k, if for U 2 n Dqa ðUÞ ¼ DðUÞDFa ðUÞ, a ¼ 1,…, k:
(7)
This is equivalent to requiring that any smooth solution U of (1) satisfies automatically the additional conservation law k X (8) @t ðUÞ + @a qa ðUÞ ¼ 0: a¼1
In the scalar case (5), anyR function (u) qualifies as entropy, with associated entropy flux qa ðuÞ ¼ 0 ðuÞf 0 ðuÞdu, a ¼ 1,…,k. Also, for k ¼ 1 and n ¼ 2, the system (7) of two equations in two unknowns, yields a rich family
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of entropy–entropy flux pairs (, q). In all other cases, however, since kn > k +1, (7) is overdetermined so one should not expect the existence of nontrivial entropies for generic systems (1). Nevertheless, it turns out that in virtually all interesting systems arising in physics, the fluxes are judiciously selected so that an entropy exists. Moreover, quite often, though not always, this entropy is a convex function of U. A case in point is the system (6) of the Euler equations, which is equipped with the entropy–entropy flux pair 1 1 (9) ¼ reðrÞ + rjvj2 , q ¼ ½reðrÞ + rjvj2 + pðrÞv, 2 2 R 2 where eðrÞ ¼ r pðrÞdr is the internal energy. It turns out that the hyperbolicity condition p0 (r) > 0 renders convex, as a function of the canonical state vector (r, rv). The extra conservation law (8) here expresses conservation of mechanical energy. Any symmetric system (1) is endowed with the entropy–entropy flux pair 1 (10) ðUÞ ¼ jUj2 , qa ¼ U Fa ðUÞ ’a ðUÞ, 2 where ’a is a ‘potential’ with D’a ¼ Fa, which exists since DFa is symmetric. It is easily seen that, conversely, any system of conservation laws (1) possessing a convex entropy (U) is symmetrized by introducing the new state vector V ¼ D(U). In particular, any such system is hyperbolic. The Cauchy problem is locally well posed for any system of conservation laws (1) endowed with a convex entropy. Indeed, for any function U0 in the 1 Sobolev space H l ðk Þ, with l > k + 1, there exists a unique C1 solution U 2 of (1) defined on k ½0, T∞ Þ and satisfying the initial condition Uðx, 0Þ ¼ U0 ðxÞ, x 2 k :
(11)
The lifespan T∞ is maximal in the sense that either T∞ ¼ ∞ or else T∞ < ∞, in which case maxk jrUð ,tÞj ! ∞, as t ! T∞ . The proof of the above proposition rests on establishing a priori bounds on the L2 ðk Þ norms of U and all of its spatial derivatives up to order l. These are derived by means of ‘energy’ estimates induced by the presence of the convex entropy. For instance, (8) yields a bound on the L2 ðk Þ norm of U. The 1 restriction l > k + 1 is needed in order to keep k rUkL∞ bounded, in which 2 case the family of energy estimates closes. It turns out that for nonlinear hyperbolic systems of conservation laws the case of finite lifespan for smooth solutions is the rule rather than the exception. This comes as a result of the wave breaking effect: As waves move at different speeds, “compressive” wave profiles get progressively steeper and eventually break. This scenario is easily seen in the setting of the scalar conservation law. Assume u(x, t) is a local smooth solution of the Cauchy problem for (5), with initial values u(x, 0) ¼ u0(x). Consider characteristics
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associated with u as trajectories of the ordinary differential equation dxa ¼ f 0 ðuðx,tÞÞ, a ¼ 1, …, k. Letting an overdot denote the derivative dt Pa fa0 ðuÞ@a in the characteristic direction, we may write (5) as u_ ¼ 0, @t + which shows that characteristics are straight lines along which u is constant. Thus, with any (x, t) is associated y 2 k such that ya ¼ xa tfa0 ðuðx,tÞÞ, a ¼ 1, …,k
(12)
and u(x, t) ¼ u0(y). This easily implies @b uðx,tÞ ¼ P
@b u0 ðyÞ , k X 0 1 + t @a fa ðu0 ðyÞÞ a¼1
b ¼ 1, …,k:
(13)
@a fa0 ðu0 ðyÞÞ takes negative values at any points, 1 attaining a negative minimum, say e, on k , ru must blow up at t ¼ . e Wave breaking is particularly pronounced in one space dimension, where waves are confined and cannot avoid interacting with one another. In several space dimensions, depending on the geometry of the system, wave breaking may be impeded by wave dispersion, which has the opposite effect. As we saw earlier, this is not the case for the scalar conservation law (5). The situation is quite different for the Euler equations (6), in three spatial dimensions. For this system, wave breaking and dispersion are evenly matched and their competition is very keen. Dispersion manages to prolong thelifespan of 1 smooth solutions with initial derivatives of size e to O exp —much longer e 1 lifespan of smooth solutions to the scalar conservation law. than the O e Nevertheless, eventually wave breaking prevails and the derivatives of the solution blow up. In view of the breakdown of smooth solutions, in order to get solutions in the large, one has to resort to weak, distributional solutions, namely bounded measurable vector fields U on some domain X of k , which satisfy Z Z k X (14) ½@t fU + @a fFa ðUÞdxdt ¼ 0, We conclude that if
X
a¼1
for all smooth test functions f with compact support in X . Notice that it is possible to define weak solutions because conservation laws (1) are in divergence form. In particular, let us seek weak solutions defined on all of k in the form 8 < U for x st < 0 Uðx, tÞ ¼ (15) : U + for x st > 0,
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where U, U+ are constant states in n , 2 k1 and s is a scalar. It is a simple exercise to verify that U in (15) will satisfy (14) for all test functions f if and only if k X (16) a ½Fa ðU + Þ Fa ðU Þ s½U + U ¼ 0: a¼1
Recalling the definition of hyperbolicity one sees that for any fixed U 2 n and 2 k1 it is possible to find U+ in the vicinity of U such that U+ U is nearly collinear to Ri(U, ) and (16) holds for some s near li(U, ). Such a weak solution of (1) is termed a planar shock wave with amplitude U+ U propagating in the direction with speed s, and (16) is called the Rankine–Hugoniot jump condition. More generally, there exist piecewise smooth weak solutions to (1) with jump discontinuities across curved shocks. In that situation (16) still holds across the shock, though now , U, U+ and s are no longer constant, as they may vary from point to point on the shock. Familiarity with weak solutions to hyperbolic conservation laws is enhanced by visualizing them as composites of continuous waves and shock waves, with the understanding that these two constituents may be finely blended. Though it is not presently known whether every L∞ weak solution fits the above description, this is certainly the case at least for solutions U of class BV, for which @ tU and @ aU are Radon measures. Indeed, the domain of U 2 BV is the union of three disjoint sets, namely: (a) the set of points of approximate continuity, in the sense of Lebesgue; (b) the set of points of (approximate) jump discontinuity, which is an at most countable family of disjoint C1k-dimensional manifolds, across which the jump condition (16) holds; and (c) a ‘small’ residual set whose k-dimensional Hausdorff measure is nil. As we shall see in the next section, a serious obstacle for dealing with weak solutions is the loss of uniqueness in the Cauchy problem. Accordingly, additional requirements must be imposed, in order to weed out spurious solutions and single out the unique admissible one. In what follows, we outline two methods in that direction, which are dictated, or at least motivated, by physics. Assume our system (1) is endowed with an entropy–entropy flux pair (, q), with D2(U) positive definite. Recall that under such conditions any smooth solution of (1) satisfies automatically the extra conservation law (8). However, this is no longer the case for weak solutions of (1). We now stipulate that a bounded measurable weak solution U of (1) satisfies the entropy admissibility condition, relative to , on a domain X of k , if k X (17) @t ðUÞ + @a qa ðUÞ 0 a¼1
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holds in the sense of distributions, that is Z Z k X ½@t cðUÞ + @a cqa ðUÞdxdt 0, X
7
(18)
a¼1
for all nonnegative smooth test functions c, with compact support in X . In particular, all smooth solutions of (1) satisfy this condition and thus are deemed admissible. In the physical applications, the inequality (17) typically manifests, directly or indirectly, the second law of thermodynamics. The entropy admissibility condition is particularly effective for scalar conservation laws (5), for which, as we saw earlier, any convex function (u) may serve as entropy. The approach pioneered by Kruzkov is to deem a weak solution u of (5) admissible if it satisfies k X (19) @t ðuÞ + @a qa ðuÞ 0 a¼1
R for all convex functions (u) and qa ðuÞ ¼ 0 ðuÞf 0 ðuÞdu. It has been shown that for any u0 ðxÞ 2 L∞ ðk Þ there exists a unique admissible solution u(x, t) of (5) on k ½0, ∞Þ, with initial value u0(x). Furthermore, admissible solutions are strongly stable as they have the following L1 contraction property: Z Z juðx,tÞ uðx,tÞjdx ju0 ðxÞ u0 ðxÞjdx, 0 t < ∞, (20) k
k
holds for any pair ðu, uÞ of admissible solutions with initial values ðu0 , u0 Þ: In particular, applying (20) with u0 ðxÞ ¼ u0 ðx + eÞ and thereby uðx, tÞ ¼ uðx + e,tÞ, where e is an arbitrary k-vector, we deduce that initial data u0 of class BV generate solutions u to the Cauchy problem for the scalar conservation law (5) that are also of class BV and the variation of u(, t) over k is a nonincreasing function of t on ½0, ∞Þ. By contrast, systems of conservation laws (1) with n 2 typically possess a single convex entropy so that the admissibility condition (17) does not generally suffice for uniqueness of solutions to the Cauchy problem. In particular, it has been shown that for certain initial data the Cauchy problem for the system of the Euler equations (6) admits infinitely many weak solutions satisfying the entropy admissibility condition (17), relative to the entropy (9). Nevertheless, it turns out that even a single inequality (17) for a convex entropy suffices for securing uniqueness and stability of smooth solutions to the Cauchy problem, not only within the class of smooth solutions but even within the broader class of admissible weak solutions (so-called weak-strong tÞ, stability). Specifically, if on k ½0, T there exist a smooth solution Uðx, with initial values U0 ðxÞ, and also a weak solution U(x, t), with initial values U0(x), satisfying the entropy admissibility condition (17), then Z Z 2 jUðx, tÞ Uðx,tÞj dx ceat jU0 ðxÞ U0 ðxÞj2 dx, 0 t T: (21) k
k
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The proof of this property is established with the help of the so-called relative entropy ¼ ðUÞ ðUÞ DðUÞ½U HðU, UÞ U:
(22)
It is easy to see that for solutions of the form (15) the entropy admissibility condition (18) reduces to the jump condition k X
a ½qa ðU + Þ qa ðU Þ s½ðU + Þ ðU Þ 0:
(23)
a¼1
For arbitrary weak solutions U satisfying the entropy admissibility condition, the left-hand side of (17) is a nonpositive distribution, and thereby a measure. In particular, it turns out that when U is of class BV the above measure is concentrated on the set of points of jump discontinuity and (17) reduces to the requirement that (23) holds across every shock. This has motivated the widely held conjecture that the admissibility of weak solutions hinges exclusively on a localized test, such as (23), to be applied to every point of jump discontinuity, involving just U, U+ and s, interrelated through (16). Even though it may not be universally valid, the above premise enjoys wide applicability, as we shall see in the next section. We now turn to an alternative admissibility criterion for weak solutions, which is also motivated by physics. The isentropic flow of a viscous gas is governed by the Navier–Stokes equations 8 < @t r + divðrvÞ ¼ 0 (24) : @t ðrvÞ + divðrv vÞ + grad pðrÞ ¼ ðl + mÞgrad div v + mDv, where l and m are viscosity coefficients. Any gas in nature has some, perhaps minute, viscosity and hence admissible solutions to the Euler equations (6) should be viewed as asymptotic solutions to the Navier–Stokes equations (24), with viscosity coefficients tending to zero. One may extend the above argument to general systems (1) as follows. Next to (1), we consider the system @t U +
k X
@a Fa ðUÞ ¼ mDU,
(25)
a¼1
where m is a positive ‘viscosity’ parameter, and postulate that a weak solution U of (1) satisfies the viscosity admissibility condition if it is the m ! 0 limit of smooth solutions Um of the parabolic system (25). Assume (1) is endowed with an entropy–entropy flux pair (, q), with (U) convex. Suppose {Um} is a family of solutions of (25), with Um converging, boundedly almost everywhere, as m ! 0, to a weak solution U of (1). Multiplying (25) by D(Um) and using (7) yields
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@t ðUm Þ +
k X
k X
@a qa ðUm Þ ¼ mDðUm Þ m
a¼1
a¼1
@a Um> D2 ðUm Þ@a Um :
9
(26)
As m ! 0, the left-hand side of (26) converges, in the sense of distributions, to the left-hand side of (17). On the right-hand side of (26), the first term converges to zero while the second term stays nonpositive, since (U) is convex. We thus conclude that U satisfies (17), i.e. the viscosity admissibility condition implies the entropy admissibility condition. However, as we shall see in the next section, the converse is not generally true. In the scalar case, for consistency with the notation in (5), we write (25) in the form k X (27) @t u + @a fa ðuÞ ¼ mDu: a¼1
Solutions to the Cauchy problem for (27) have the L1 contraction property (20). This estimate serves as the tool for showing that, as m ! 0, the solution um(x, t) of the Cauchy problem for (27), with initial data u0(x) in L∞ , converges, boundedly almost everywhere, to the unique weak solution u(x, t) of (5), with the same initial values, which satisfies (19) for all convex functions (u). Thus for scalar conservation laws, the viscosity admissibility condition is equivalent to the entropy admissibility condition, for all convex entropies. By contrast, for n 2, the convergence of solutions of (25), as m ! 0, has been established only under quite restrictive hypotheses. Consequently, there is no straightforward way to test whether any particular solution of (1) satisfies the viscosity admissibility condition. In practice, the testing is performed for solutions of the form (15), with the expectation that local admissibility of shocks renders the entire solution admissible. Accordingly, one seeks to capture the solution (15) as the m ! 0 limit of solutions Um of (25) in the form x st , Um ðx,tÞ ¼ VðxÞ, x ¼ (28) m depicting fronts propagating in the direction with speed s, and becoming progressively steeper, as m decreases. Then V must satisfy the ordinary differential equation k X (29) V€ ¼ sV_ + a F_a ðVÞ a¼1
on ð∞, ∞Þ, with boundary conditions Vð∞Þ ¼ U ,Vð∞Þ ¼ U + . Integrating (29), k X (30) a ½Fa ðVÞ Fa ðU Þ: V_ ¼ s½V U + a¼1
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Notice that the right-hand side of (30) vanishes both at V ¼ U and V ¼ U+, the latter by virtue of the jump condition (16). We conclude that the solution (15) of (1) satisfies the viscosity admissibility condition if there exists an orbit of (30) joining the equilibrium points U and U+. The function V (x) is called the shock profile or the shock structure. It allows us, so to say, to observe the shock under the microscope. There are many other topics of interest in the theory of hyperbolic systems of conservation laws. When dealing with solutions of (1) on some domain of k the question of assigning proper boundary conditions requires deep analysis, even in the context of smooth solutions. One often encounters in the applications so-called systems of balance laws k X (31) @t U + @a Fa ðUÞ ¼ GðUÞ, a¼1
with a source term G(U) manifesting relaxation. In that case, the source has a dissipative effect inducing global existence of smooth solutions to the Cauchy problem, when the initial data are smooth and ‘small’. As noted earlier, the Cauchy problem for scalar conservation laws, n ¼ 1, is well-posed, in the setting of admissible weak solutions, for any k 1. On the other hand, virtually nothing is known on the existence and uniqueness of weak solutions to the Cauchy problem for systems, n 2, when k 2. However, considerable progress has been made for systems in one spatial dimension, k ¼ 1. The following section will provide an overview.
3 STRICTLY HYPERBOLIC SYSTEMS IN ONE SPATIAL DIMENSION This section surveys aspects of the theory of hyperbolic systems of conservation laws @t U + @x FðUÞ ¼ 0 (32) in one spatial dimension. As in Section 2 , the state vector U and the flux F(U) take values in n . The system is called strictly hyperbolic if for any U 2 n the Jacobian matrix DF(U) possesses n real distinct eigenvalues l1(U) < ⋯ < ln(U), called characteristic speeds, and thereby linearly independent eigenvectors R1 ðUÞ,…, Rn ðUÞ: The theory of hyperbolic systems that are not strictly hyperbolic is still incomplete, even for the simplest case where two characteristic speeds coalesce in just a single point of n . In our discussion we will employ as demonstration models the scalar conservation law @t u + @x f ðuÞ ¼ 0 and the so-called p-system
(33)
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8 < @t u @x v ¼ 0 :
(34) @t v + @x pðuÞ ¼ 0,
which p is ffiffiffiffiffiffiffiffiffiffiffiffiffi strictly whenffi p0 (u) < 0, with characteristic speeds ffi hyperbolic pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 l1 ¼ p ðuÞ and l2 ¼ p0 ðuÞ. The p-system is the Lagrangian version of the Euler equations (6), in one spatial dimension. It governs the rectilinear flow of a gas in a duct, with u standing for specific volume (the inverse of density) and v denoting velocity. The same system governs the one-dimensional motion of elastic solids (longitudinal vibrations of a bar, shearing motion of a slab, oscillation of a string, etc.). In that context one usually replaces in (34) the pressure p with the negative stress s. The contrast in behaviour between linear and nonlinear hyperbolic systems (32), in one spatial dimension, is particularly pronounced when the latter satisfy the genuine nonlinearity condition, namely, after normalizing the eigenvectors Ri, (35) Dli ðUÞRi ðUÞ ¼ 1, i ¼ 1, …, n, U 2 n : In particular, the scalar conservation law (33) is genuinely nonlinear when f 00 (u) 6¼ 0, for all u. The simplest example is the celebrated Burgers equation 1 @t u + @x u2 ¼ 0, (36) 2 which is ubiquitous in the area of conservation laws. Despite its apparent simplicity, (36) exemplifies many of the principal features of genuinely nonlinear systems of hyperbolic conservation laws and provides an excellent model for an initial approach to the subject. The p-system (34) is genuinely nonlinear when p00 (u) 6¼ 0, for all u. This is the natural assumption for fluids, where p00 (u) > 0, while in solids the stress function s(u) may have inflection points. As we saw in the previous section, smooth solutions to the Cauchy problem for the scalar conservation law, in any number of spatial dimensions, typically break down in finite time because of wave breaking. This process is particularly transparent in the setting of the Burgers equation. Assume u is a smooth solution to the Cauchy problem for (36), defined on dx ð∞, ∞Þ ½0, TÞ. Consider the characteristics ¼ uðx, tÞ and denote differdt entiation @ t + u@ x in the characteristic direction by an overdot. Differentiation of (36) with respect to x yields that the derivative v ¼ @ xu satisfies the Bernoulli equation v_ + v2 ¼ 0, along the characteristic. Thus if v(0) ¼ e < 0, 1 @ xu must blow up at time t ¼ . The same mechanism is responsible for the e blowing up of smooth solutions to the Cauchy problem for general genuinely nonlinear hyperbolic systems of conservation laws in one spatial dimension,
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but the proof is considerably more complicated. One should expect generic blowing up of smooth solutions even for nonlinear hyperbolic systems of conservation laws that are not genuinely nonlinear, but this needs to be demonstrated individually for each case. In view of the above, to solve the Cauchy problem in the large one has to resort to weak solutions that contain shocks. The Rankine–Hugoniot jump condition (16), satisfied across shocks, here reduces to FðU + Þ FðU Þ s½U + U ¼ 0:
(37)
The standard terminology is that ‘U+ is joined to U by a shock of speed s’. In particular, for the scalar conservation law (33), s¼
f ðu + Þ f ðu Þ : u + u
(38)
We now demonstrate, in the context of the Burgers equation, the loss of uniqueness of weak solutions, reported in Section 2. Indeed, the Cauchy problem for (36) with initial data u(x, 0) ¼ 1, for x < 0, and u(x, 0) ¼ 1, for x > 0, admits infinitely many weak solutions, including the following two: 8 x > 1 ∞ < < 1 > > t < x x 1 1 (39) uðx, tÞ ¼ > t t > x > : 1 1 < < ∞, t 8 x < 1 ∞ < < 0 t uðx,tÞ ¼ (40) x : 1 0 < < ∞: t This pathology is encountered in every nonlinear system (32) of conservation laws. In order to weed out spurious solutions, so as to restore uniqueness to the Cauchy problem, we appeal to the admissibility criteria introduced in Section 2, as related to shocks. We begin with the case of the scalar conservation law. Consider a shock for (33) joining the states u and u+. Its speed s is given by (38). Eq. (30) for the structure of the shock here takes the form v_ ¼ sðv u Þ + f ðvÞ f ðu Þ:
(41)
The shock will satisfy the viscosity admissibility condition if there exists a solution v(t) of (41) on ð∞, ∞Þ, with vð∞Þ ¼ u and vð∞Þ ¼ u + . Thus the right-hand side of (41) should not change sign between u and u+, or equivalently 8 < 0 if u < v < u + f ðvÞ hðvÞ (42) : 0 if u + < v < u ,
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where h(v) is the straight line segment (chord), with slope s, connecting the points (u, f(u)) and (u+, f(u+)) on the graph of f. This is the celebrated Oleinik E-condition. Looking at the same question from the perspective of the entropy admissibility condition, we fix any convex R function (u) as entropy and determine the associated entropy flux qðuÞ ¼ 0 ðuÞf 0 ðuÞdu: After a simple calculation, (23) here yields Z u+ 00 ðvÞ½ f ðvÞ hðvÞdv ¼ qðu + Þ qðu Þ s½ðu + Þ ðu Þ 0: (43) u
It is clear that the Oleinik E-condition (42) is equivalent to (43), for all convex (u). On the other hand, (43) for just one convex (u) does not necessarily imply (42), unless f 00 (u) 6¼ 0, for all u, i.e. when (33) is genuinely nonlinear. In the genuinely nonlinear case, Eqs. (42) and (43) reduce to the celebrated Lax E-condition f 0 ðu + Þ < s < f 0 ðu Þ:
(44)
In particular, for the Burgers equation (36), the Lax E-condition (44) reduces to u+ < u, so that the solution (40) violates the viscosity and the entropy admissibility conditions. In fact (39) is the unique admissible solution to that Cauchy problem. We now turn to the question of admissibility of shocks for general strictly hyperbolic systems of conservation laws (32). The first task is to determine the Hugoniot locus, associated with a fixed state U, which consists of all states U+ that satisfy (37) for some s, and thereby may be joined to U by a shock. The part of the Hugoniot locus contained in a small neighbourhood of U has a definite and simple structure. Notice that when (37) holds with jU+ Uj small, then s must be close to one of the characteristic speeds li(U) and U+ U must be almost collinear to the associated eigenvector Ri(U). Indeed, since (32) is assumed to be strictly hyperbolic, it follows by standard bifurcation theory that for each i ¼ 1,…, n there exists a smooth curve Wi(t) in some neighbourhood of U and associated scalar function si(t) such that FðWi ðtÞÞ FðU Þ si ðtÞ½Wi ðtÞ U ¼ 0,
(45)
Wi ð0Þ ¼ U ,
(46)
W_ i ð0Þ ¼ Ri ðU Þ,
(47)
si ð0Þ ¼ li ðU Þ,
(48)
1 s_i ð0Þ ¼ Dli ðU ÞRi ðU Þ: 2
(49)
Wi(t) is called the i-shock curve of (1) through U and the Hugoniot locus near U is the union of the n shock curves.
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As an example, consider the p-system (34), with Rankine–Hugoniot conditions (37) in the form 8 < v + v + sðu + u Þ ¼ 0 (50) : pðu + Þ pðu Þ sðv + v Þ ¼ 0, whence
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðu + Þ pðu Þ : s¼ u + u
(51)
It is here convenient to parametrize the shock curve through (u, v) by u, in which case v ¼ v sðu u Þ, (52) taking s from (51), with the negative square root for the 1-shock curve and the positive square root for the 2-shock curve. For the above simple system, (52) describes not only the local but even the global Hugoniot locus. In general, however, the global portrait of the Hugoniot locus may be quite complex, containing detached branches, islands, etc. When the system is not strictly hyperbolic, even the local portrait of the Hugoniot locus may be geometrically varied and complex. Accordingly, here we shall limit discussion to systems (32) that are strictly hyperbolic and for shocks of small amplitude. The i-shocks must pass the admissibility test. For the viscosity admissibility condition, (30) here reduces to V_ ¼ s½V U + FðVÞ FðU Þ:
(53)
One seeks a solution V(t) of the ordinary differential equation (53) on ð∞, ∞Þ, with boundary conditions Vð∞Þ ¼ U , Vð∞Þ ¼ U + . On the other hand, if (32) is endowed with an entropy–entropy flux pair (, q), with (U) convex, the entropy admissibility condition (23) for the shock here reduces to qðU + Þ qðU Þ s½ðU + Þ ðU Þ 0:
(54)
As in the scalar case, it turns out that the viscosity condition always implies the entropy condition. Furthermore, whenever the system (32) is genuinely nonlinear (35), then an i-shock of small amplitude satisfies the viscosity condition if and only if it satisfies the entropy condition, and also if and only if the Lax E-condition li ðU + Þ < s < li ðU Þ (55) holds. Thus, with reference to the i-shock curve and by virtue of (48) and (49), we deduce that, for genuinely nonlinear systems, an i-shock with small amplitude will be admissible when U+ ¼ Wi(t), for t < 0, and inadmissible when U+ ¼ Wi(t), for t > 0.
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When the system (32) is not genuinely nonlinear, the viscosity admissibility condition still manages to weed out all inadmissible shocks, whereas the entropy condition and the Lax E-condition are less selective, being merely necessary but not sufficient for admissibility. A strengthening of the Lax E-condition that is equivalent to the viscosity condition, even when the system is not genuinely nonlinear, is provided by the Liu E-condition, which stipulates that an i-shock joining U+ ¼ Wi(t) to U will be admissible if si(s) si(t), for all s between 0 and t. Thus, the Liu E-condition generalizes the Oleinik E-condition, from scalar conservation laws to strictly hyperbolic systems of conservation laws. The issue of admissibility of shocks in systems that are not strictly hyperbolic is still unresolved. An important feature of systems (32) is that they are invariant under uniform stretching of the space-time variables, as a result of which they admit self-similar solutions in the form x Uðx,tÞ ¼ VðxÞ, x ¼ : (56) t is a trivial case of a self-similar solution. Any constant state, VðxÞ ¼ U, Another important example is provided by step functions, V(x) ¼ U, for x < s, and V(x) ¼ U+, for x > s, where U and U+ satisfy (37) and thus may be joined by a shock of speed s. Next we consider the possibility of Lipschitz continuous self-similar solutions (56), with V(x) ¼ U, for x x, V(x) ¼ U+, for x x+, and V(x) smooth, for x x x+. In that case, one says that ‘U+ is joined to U by a centred rarefaction wave’—terminology borrowed from gas dynamics. We investigate this question under the assumption that the system (32) is genuinely nonlinear (35). Substituting from (56) into (32) yields ½DFðVðxÞÞ xIV_ ðxÞ ¼ 0:
(57)
li ðVðxÞÞ ¼ x,
(58)
Thus, assuming V_ ðxÞ 6¼ 0, and V_ ðxÞ is collinear to Ri(V (x)), for some i ¼ 1,…, n (i-rarefaction wave). Differentiating (58) with respect to x and using (35) yields V_ ¼ Ri ðVÞ:
(59)
Thus x ¼ li(U), x+ ¼ li(U+) and U+ ¼ V(x+), where V(x) is the solution of the ordinary differential equation (59) with initial value V(x) ¼ U. We reparametrize V(x), replacing x with the new parameter t ¼ xx, denote it by Vi(t) and call it the i-rarefaction curve through U. As an example, consider the p-system (34), under the assumption p00 (u) > 0 of genuine nonlinearity. As with the shock curve (52), it is convenient to parametrize the rarefaction curve through (u, v) by u, in which case it assumes the form
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Z u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ðwÞdw, v ¼ v
(60)
u
where the minus sign gives the 1-rarefaction curve and the plus sign yields the 2-rarefaction curve. Fig. 1 provides a sketch of both the i-shock curve and the i-rarefaction curve through U. Notice that if we discard the part of the shock curve associated with inadmissible shocks and concatenate the admissible part of the i-shock curve with the i-rarefaction curve, we end up with a C1 curve called the i-wave curve through U. The celebrated Riemann Problem is the Cauchy problem for (32) with initial data ∞ < x < 0 Ul (61) Uðx, 0Þ ¼ Ur 0 < x < ∞, where Ul and Ur are given constant states. Since the above initial data are invariant under stretching of the x-variable, one expects that the solution to the Riemann Problem will be self-similar, (56). As (32) remains invariant under extreme stretches and extreme contractions of the space-time variables, it is expected that the solution of the Riemann problem should depict both the local and the large time behaviour of admissible weak solutions to the Cauchy problem, under general initial conditions. One might say that the solution of the Riemann problem provides the instrument for observing general solutions ‘under the microscope’ as well as ‘through the telescope’. Furthermore, the solution to the Riemann problem has been used as a building block for constructing general solutions, for theoretical or numerical purposes. For genuinely nonlinear systems, it is possible to synthesize the solution to the Riemann problem with the help of the wave curves, introduced above. Fig. 2 shows how this is done for the case of the p-system (34). Given the endstates (ul, vl) and (ur, vr), one has to locate an ‘intermediate’ state (um, vm) with the property that (um, vm) lies on the 1-wave curve through (ul, vl) and, in turn, (ur, vr) lies on the 2-wave curve through (um, vm). This is possible because 1- and 2-wave curves intersect transversely, by strict hyperbolicity. The resulting solution to the Riemann Problem consists of the three constant states (ul, vl), (um, vm), (ur, vr) together with a 1-wave joining (um, vm) to
FIG. 1 Shock and rarefaction wave curves.
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A
B
FIG. 2 Solution to the Riemann problem.
(ul, vl) and a 2-wave joining (ur, vr) to (um, vm). Because of the relative location of (ul, vl), (ur, vr) in Fig. 2 , the 1-wave turns out to be a shock and the 2-wave a rarefaction. However, any combination is possible. The situation is similar for general genuinely nonlinear systems (32). By means of the implicit function theorem, one shows that for any given states Ul and Ur, with jUr Ulj sufficiently small, there exists a unique solution to the Riemann problem, with initial data (61), which is composed of n + 1 constant states Ul ¼ U0, U1,…, Un ¼ Ur, with Ui joined to Ui1 by either an admissible i-shock or by an i-rarefaction wave. The solution to the Riemann problem for strictly hyperbolic systems (32) that are not genuinely nonlinear has a similar structure, involving as above n +1 constant states Ul ¼ U0, U1,…, Un ¼ Ur. However, now Ui is joined to Ui1 by an i-wave that is no longer necessarily a single i-shock or i-rarefaction wave, but possibly a composite of several such shocks and rarefactions. For systems that are not strictly hyperbolic, the solution to the Riemann problem may assume a variety of forms and the issue of admissibility and uniqueness is not completely resolved. We now turn to the general Cauchy problem for strictly hyperbolic systems (32), under initial conditions Uðx,0Þ ¼ U0 ðxÞ, ∞ < x < ∞:
(62)
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The state of the art may be summarized as follows: If the total variation of U0(x) is sufficiently small, there exists a unique admissible weak solution U(x, t) of (32), (62) on the upper half-plane. The function U is in BV with respect to the space-time variables. Furthermore, for each fixed t, the function U(, t) has bounded variation on ð∞, ∞Þ and TVð∞, ∞Þ Uð , tÞ cTVð∞, ∞Þ U0 ð Þ,
0 t < ∞:
(63)
The above solution has been constructed by three methods, namely: (a) The random choice method of Glimm, with essential extensions by Tai-Ping Liu. It uses solutions to the Riemann problem as building blocks for approximating the general solution. (b) The front tracking algorithm developed by the Italian School headed by Bressan. The strategy in this approach is to construct approximate solutions that contain only constant states and shocks, by replacing rarefaction waves with fans of inadmissible, albeit very weak, rarefaction shocks. Shock interactions are then handled by solving Riemann problems. (c) The vanishing viscosity approach of Bianchini and Bressan. In all three methods, the crucial step is to establish the bound (63) on the approximate solutions, for some uniform constant c. For the above theorem to hold, the restriction that the total variation of the initial data must be small is essential. Indeed, there are cases of systems in which weak solutions under initial data with large total variation break down in finite time. A major challenge to the theory at the present time is to determine whether the systems of hyperbolic conservation laws arising in physics are free from such pathologies. An even greater challenge is the issue of existence of weak solutions in the large, for systems in more than one spatial dimension: This is currently terra incognita.
REFERENCES Bressan, A., 2000. Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford University Press, Oxford. Dafermos, C.M., 2016. Hyperbolic Conservation Laws in Continuum Physics, fourth ed. Springer, Heidelberg. Holden, H., Risebro, N.H., 2015. Front Tracking for Hyperbolic Conservation Laws, second ed. Springer, New York. Serre, D., 1999. Systems of Conservation Laws, vols. 1–2. Cambridge University Press, Cambridge. Smoller, J.A., 1994. Shock Waves and Reaction-Diffusion Equations, second ed. Springer, New York.