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6 The Cauchy Problem for Systems in Several Dimensions
6
The Cauchy Problem for Systems in Several Dimensions
The results of Chapter 3 about linear parabolic, strongly hyperbolic, and mixed systems can mostly be generalized from one space dimension to an arbitrary number of space dimensions. These generalizations are carried out in the first three sections of this chapter. In particular, we show well-posedness of the Cauchy problem for the linearized compressible Navier-Stokes and Euler equations. Section 6.4 treats short-time existence for nonlinear systems; we carry out the details for the symmetric hyperbolic case and sketch generalizations to systems of different type. For a special class of nonlinear parabolic systems in two space dimensions, we will prove all-time existence in Section 6.5.
6.1. Linear Parabolic Systems We consider second-order parabolic systems of the general form (6.1.1)
ut =
2
I.J=i
Dt(A,,D,u)
+
2
B,D,ir
+ Cu + F.
,=I
Here s is the number of space dimensions and D , is the operator D, = i)/ds, . The coefficient matrices A,, = A,,(z,t), €3, = B , ( z , t ) , C = C ( s . t )and the forcing function F = F ( z ,t ) are assumed to be C”-smooth, real, and 1-periodic
I77
178
Initial-Boundary Value Problems and the Navier-Stokes Equations
in each component xi,i = 1 , . .. , s. The concept of parabolicity will be defined below. For the function u(x,t) an initial condition (6.1.2)
u(z:O)= f(x), f E C",
f I-periodic in each xi
is given, where f takes values in R". As before, we seek a smooth solution u = u(x, t) taking values in R" which is 1-periodic in each xi, i = 1 , . . . , s. In Section 6.1.1 the concept of strong parabolicity will be defined; it leads to a simple derivation of solution-estimates. Existence of a solution can again be shown by proving the analogous estimates for a difference scheme. This will be sketched in Section 6.1.2. Existence, uniqueness, and estimates can be generalized from strongly parabolic systems to systems for which a pointwise definition of parabolicity is used. For our applications this generalization is of minor importance, however, and we only sketch the proof.
6.1.1. Estimates for Strongly Parabolic Systems The following definition generalizes our previous concept of strong parabolicity to systems in any number of space dimensions.
Dejnition. The system (6.1.1) is called strongly parabolic in 0 5 t 5 T if there exists a constant 6 > 0 such that the estimate
i= I
holds for all x E R", 0 5 t 5 T, and all vectors yi E R", i = 1 , . . . , s. The basic result about existence, uniqueness, and estimates is
Theorem 6.1.1. If(6.1.1) is stronglyparabolic in 0 5 t 5 T , then the Cauchy problem (6.1. I ) , (6.1.2) has a unique classical solution u = u(x,t ) in 0 5 t 5 T . The solution u is a Cm-function. For each p = 0 , 1,2,. . ., there is a constant K p with
The constant K p depends on 6 and the maximum norm of the coefficients of (6.1.1) and their derivatives of order 5 p , but not on F, f.
The Cauchy Problem for Systems in Several Dimensions
179
Proof. We start with the estimates. Integration by parts yields I d 2 dt
- - ( u , u ) = ( u ,u t ) S
S
Therefore, (6.1.3) follows for p = 0 if we apply Lemma 3.1.1. This estimate also shows the uniqueness of a classical solution. To estimate first derivatives, we differentiate the equation (6.1.1) with respect to z,, i = 1 , .. , , s , and obtain a system of similar type for the vector
(DIU,D2u,. .. . D,u). The coefficients of this system depend on A , j , B,, C and the first derivatives of these matrices. The desired estimate follows as above. Estimates for higher derivatives are obtained by repeated differentiation.
In the next section we sketch the proof of the existence of
u.
6.1.2. Existence via Difference Approximations Let h = 1/N denote a gridlength, N a natural number. For any multi-index
v = (v1... . ,vs), vj E z. let z, = (hvl , . . . , h ~ , denote ~ ) the corresponding meshpoint. We consider gridfunctions v, = v(z,,) which are assumed to be 1-periodic in each coordinate direction. By EJ we denote the translation operator in the j-th coordinate:
Ejuv = Ejv(z,)= u ( z ,
+ he,),
e3 = (0.... , 0 , 1 , 0 , . . . , 0) with 1 at position j The powers of EJ are
EJPu,, = v(s,
+ phe,),
p E Z,
and EJ”= I is the identity. The forward, backward, and centered difference operators in the 3-th coordinate direction are
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Initial-Boundary Value Problems and the Navier-Stokes Equations
1 2
DOj = - (D+j
+ D-j),
j = 1 , . .., s .
Using these notations, we approximate (6.1.1) by the system of ordinary differential equations
i=l
If we introduce a discrete scalar product and norm for gridfunctions by
then all computational rules of Section 3.2.2 carry over in an obvious way to the multi-dimensional case. Therefore, we can estimate the solutions of (6.1.4) and their divided differences independently of h. We just mimic the continuous process. As in the one dimensional case, we can interpolate v,(t) = v,h(t)with respect to the z-variables by Fourier polynomials w"(z, t). (See Appendix 2.) These are uniformly (with respect to h) smooth; in the limit h + 0 we obtain a solution u = u ( z ,t ) of the Cauchy problem. There is no essential difference from the one dimensional case. 6.1.3.
General Linear Second-Order Parabolic Systems
In Section 2.4.3 we have defined parabolicity for problems with constant coefficients. Applying this concept to all frozen-coefficient problems, we are lead to: Definition. The system (6.1.1) is called parabolic in 0 5 t 5 T ifthere is a constant 6 > 0 such that for all x E R",0 5 t 5 T , and all w = (wI, . . . ,w,)E Rs the eigenvalues K of the matrix I
satisfy Re
K
2 61wI2.
The Cauchy Problem for Systems in Several Dimensions
181
One can prove
Theorem 6.1.2. The statements of Theorem 6.1.1 are valid if the assumption of strong parabolicity is replaced by the assumption of parabolicity. We only sketch the proof of this result. The essential point is to obtain the a priori estimates (6.1.3); then existence of a solution follows as before by use of a difference scheme. To obtain the estimate (6.1.3) for p = 0, first fix a point (so, t o ) and consider the constant-coefficient parabolic operator s
~ 2 ( 5 n t, o , D )
=
C 1
D , ALJ(so, t o ) D,.
]=I
If we proceed as in Section 2.7 - but replace Fourier transforms by Fourier expansions - we can construct an inner product (u,
to) =
C(a(u), ~ ( z oto, , ~)s(w)) W
with
(. P2(zo. to. D)u)H(c",t") + (p2(zo,to. O u , . ) H ( s o , t r , ) <- -6
c 5
llDL4*
?=I
for all smooth functions u = u(z). Using the corresponding norms and a partition of unity argument as in Section 3.2.5, we obtain the basic energy estimate as before. Estimates for derivatives can be obtained by differentiating the differential equation. As noted in Section 3.2.6, a basic property of parabolic differential operators is their smoothing ability. This smoothing property is also valid for equations in more than one space dimension and can be proved as in the one dimensional case
6.2. Linear Hyperbolic Systems A first-order system
(6.2. la)
ut
= P ( z ,t , d/dz)u
+ C(a,t)u + F ( z .t )
is called symmetric hyperbolic if P ( z . t . d / d z ) has the form (6.2. I b )
c{
1 "
P ( z ,t , d / d x ) = ~2 ,=I
B 3 ( r ,t)D,u
+ D3(BJ(x, t)~)},
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Initial-Boundary Value Problems and the Navier-Stokes Equations
and B j ( x ,t ) = Bj+(z,t ) for all x , t. As before, the coefficients B j , C , and the forcing function F are assumed to be real, Cm-smooth, and 1-periodic in each Xj.
To obtain existence for the Cauchy problem (6.2. l), (6.1.2), we consider first the parabolic problems ut = t AU + P ( x , t , d / d x ) u + CU+ F,
t
> 0.
The existence result of Theorem 6.1.1 applies, and the estimates are independent of E > 0. For E -+ 0, one obtains a solution of the symmetric hyperbolic problem. A basic property of hyperbolic equations is the “finite speed of propagation”. We formulate the result in the following theorem; the proof uses the convergence of an explicit difference scheme.
Theorem 6.2.1.
Define
The solution-value u(x0,to) depends only on those values of F ( x ,t ) for lying in the cone
1x0 - 21 and only on those values off
I a(to - t ) ,
( 2 )for
05t
( 2 ,t )
I to,
x lying in the interval 1x0 - x(5 ato.
h I
FIGURE 6.2.1. Domain of dependence and mesh.
X
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The Cauchy Problem for Systems in Several Dimensions
Proof. 1. As in Section 6.2.1, we introduce meshpoints x u , v E Z". but discretize in time also. Let k > 0 denote the time step-size. We approximate (6.2.1) by the difference approximation
( I - k C ( z u ,t ) ) li,(t
+ k ) = ( I + kC(x,, t ) ) v u ( t - k ) + 2kQ(xu, t ) v,At) + 2 k F ( x u ,t ) ,
(6.2.2)
t
= k . 2k. 3 k , . . . ,
Q(xu,t ) t ~ u ( t = )
" 51 C { B j ( r , , t)Doju,(t) + Do] ( B j ( z u ,t)v,(t))} . )=I
Here 1 Doj = -(D+, 2
+ D - j ) = -2h1( E j
I
- Ej- )
denotes the usual centered difference operator. The three level difference scheme (6.2.2) is supplemented by the starting conditions
(6.2.3) ~ ~ (=0f ( )x u ) , v u ( k )= f ( z u ) k { Q(zu.0)
+
+ C(s,. O)}f(x,) + k F ( x U 0). .
+
We fix a relation k = h / ( a q ) , 77 > 0 arbitrary, and send h + 0. The difference equations (6.2.2), (6.2.3) define an approximation t i = uh for the exact solution u, and we will prove the convergence
The domain of dependence of uh belongs to the cone 1x0 - 21
I ( a + q)(to - t ) , 0 I t
I to.
Since q > 0 is arbitrary and u is smooth, the assertion of the theorem follows. 2. It remains to show (6.2.4). First note that the operators Doi are antisymmetric:
184
Initial-Boundary Value Problems and the Navier-Stokes Equations
We give a simple bound for Q ( t ) : The estimate
implies that
L
j=l
The smooth solution u = u ( z ,t ) of the differential problem satisfies
Thus we can consider u,(t) as a solution of perturbed difference equations:
Here $,(t), 9, are bounded independently of h and k . The difference solves
Taking the inner product with m,(t find that
+ k ) + w,(t
-
UJ= u--v
k ) and summing over v, we
The Cauchy Problem for Systems in Several Dimensions
185
This implies a lower bound for L(t),
and therefore the above recursive estimate gives us that
+
U t + k ) I L ( t ) kc&t
+ k ) + kc,L(t) + k(k2 + h 2 ) 2 C 4 .
Hence the auxiliary quantities L ( t ) also satisfy
L(t + k ) 5 ( I
+ k q ) L ( t )+ k(k2 + h 2 ) 2 C 6 .
Initially, at t = k , we have L ( k ) = llw(k)11; = O(k4), and by a discrete analog of Gronwall's Lemma (Lemma 3.1.1) it follows that
L ( t ) = O((k2 + h 2 ) 2 ) , 0 5 t 5 to. Since L(t) bounds
Ilci~(t)ll;~ , the
convergence (6.2.4) is proved.
186
Initial-Boundary Value Problems and the Navier-Stokes Equations
The existence result mentioned above for symmetric hyperbolic systems can be extended to more general hyperbolic equations. Consider the first-order system (6.2.la) where the operator P ( z . t,a/az)has the form (6.2.1b), but the matrices B,(z, t) are not necessarily symmetric. We call s
P(z,t,iw) =
iC
BJ(z,t)wj. w E R", IwI = 1.
J=l
the symbol of the differential operator P and make the The system (6.2.1) is called strongly hyperbolic if there exists a smooth 1-periodic symmetrizer; i.e., there is a 1-periodic positive definite Hermitian matrix function
Definition.
H ( x , t , w ) , z E R". t 2 0, w E R", lwl = 1 , depending smoothly on all variables, with
+
H ( r . t , w ) P ( z ,t , iw) P*(Z,t , i w ) H ( z .t , w)= 0. This definition generalizes the conditions discussed in Section 2.4.1 for the multi-dimensional constant-coefficient case and the conditions in Section 3.3.1 for the one dimensional variable-coefficient case, but it requires in addition smooth dependence on (2, t. w). (In the constant-coefficientcase smooth dependence on w was not essential.) If we require that all frozen-coefficient problems obtained from (6.2.1) be strongly hyperbolic in the sense of Section 2.4.1, then we can use the matrices
S ( s . t . w ) , w E R". IwI = 1 . of Theorem 2.4.1 and obtain the symmetrizer H = S'S; compare Lemma 2.4.2. The only extra assumption in the above definition is the requirement of periodicity and smoothness of H = H ( z . t , w ) . The matrix S - ' ( x , t , w ) contains in its columns the eigenvectors of P ( z .t , iw). Thus, if the eigenvalues of P ( r ,t. iw)are all purely imaginary and if linearly independent eigenvectors can be chosen as smooth periodic functions of ( z , t . w ) , then the system is strongly hyperbolic. This is the case if the system is strictly hyperbolic, i.e., the eigenvalues of P ( z ,t , iw)are purely imaginary and always distinct. One can prove
Theorem 6.2.2. If the system (6.2.1) is strongly hyperbolic, then the Cauchy problem is well-posed. For every set of smooth data F = F ( z . t ) , f = f(x)
The Cauchy Problem for Systems in Several Dimensions
187
there exists a unique smooth solution. Estimates as in the one dimensional case are valid; i.e., the solution is as smooth as the data. Sketch of the proof. We use the theory of pseudodifferential operators and refer to Eskin (1981, Chapter 5) or Nirenberg (1973). To conform with the usual framework, we do not assume that the data and coefficients are 1-periodic. Instead, we assume that F, f E C" and that the coefficients are constant outside a bounded set. Because of the finite speed of propagation, this is no restriction. Let 4j = 4j(T) denote a partition of unity where the support of 4j has diameter dJ and sup, d3 is sufficiently small. Also, at every point T at most p functions #3 are different from zero, p independent of T . We define the pseudodifferential operator H(t) by
For sufficiently small d j
defines a scalar product which is equivalent to the usual Lz-scalar product. To prove this, we define
H,(t)v(c)= and note that
L.
H ( z , , t , W / I C Ja ( ( w) ) dP w , (xJ ~E. ~ ) 4,, support
This ensures that
can be estimated from below by const 1 1 ~ 1 1 . Let 4 = $(lwl),# E C", denote a "cut-off' function with
We write (at each t )
H=H+HR?
Initial-Boundary Value Problems and the Navier-Stokes Equations
188
where
6(w)dw. With these notations,
+ Re ( u j ,HPuj) = (ujl (HP + P*H*)uj + Re ( q HPuj). , 2
Re ( u j ,HPuj) = Re (uj ,HPuj)
Again, HP + P*H*is a pseudodifferential operator. The calculus of pseudodifferential operators implies that its symbol has the form
4(lwl) (m? w/lwl)P(z,t , iw)+ P*(X,t , i w ) H ( z ,t ,w/lwl)) t l
+ S(Z, t , w).
+
The part multiplied by 4 vanishes, and S(z, t , w) is bounded. Hence, HP P*H*is a bounded operator. Because the factor 1 - 4(lwl) in the definition of H cuts off high frequencies, the operator HP *isalso bounded. Therefore one obtains that d (u, dt u ) H ( t ) 5 const ( u !U ) H ( t )
+
(u1
F)H(t):
and the basic energy estimate follows.
6.3. Mixed Hyperbolic-Parabolic Systems and the Linearized Navier-Stokes Equations There are no problems in obtaining existence and uniqueness results, and estimates of smooth solutions for mixed hyperbolic-parabolic systems of the form
~t = P I ( z t, , d
+
+
/ d ~ ) v R ~ [ (tX , d, / d ~ )G~I .
Here we assume that the uncoupled systems (6.3.2)
+ G, Vt = P~(x, t ,d / d z ) T + F
Tit = P ~ ( x t ,,d / d ~ ) i i
are second-order strongly parabolic and first-order symmetric hyperbolic, respectively. The coupling terms R12 and R ~ are I general first-order operators.
189
The Cauchy Problem for Systems in Several Dimensions
The proof proceeds as in the one dimensional case; we refer to Section 3.4.1. The solutions are as smooth as the data. One can also generalize this result to the case where the uncoupled systems (6.3.2) are second-order parabolic and first-order strongly hyperbolic, respectively. The proof for this more general case requires that one modify the Lznorm to obtain the estimates. We shall now discuss how these results apply to the linearized Navier-Stokes equations. The (nonlinear) viscous compressible system reads p
(6.3.3)
D -u Dt
+ grad p = p A u + ( p + p ’ ) grad (div u ) + pF,
D Dt
+ p div u = 0,
-p
p = r(p),
D - d d d d - - + u - + 2) - + w -. Dt dt dy dz dx
-
Suppose U(x, t ) . P(x,t ) , R(x,t ) are smooth functions with R > 0 and P = r(R); these functions are considered as an exact or approximate solution of (6.3.3). We substitute
u=U+U’,
p=P+p’,
p=R+p’
into (6.3.3) and neglect all terms which are quadratic in the corrections u’, p’. p’. If we drop the ’ sign in our notation, we obtain the linear equations
D RTu Dt
D
(6.3.4)
-
Dt
D + p Dt U = pAu + ( p + p’) grad (div u) + G I
+
K
grad p
D
Dt
p
grad)U
d = at-a + u -+v -d + w K
-
+ R(u.
7
d dz’
-
dr
= -(R),
dP
+ R div u + u . grad R + p div U = G2.
The inhomogeneous terms G I , G2 are determined by U , P, R and vanish if U, P, R solve (6.3.3) exactly. Clearly, the equations (6.3.4) are of the general form (6.3.1) with u = u, 2) = p . For p > 0, p’ 2 0, the second-order operator
pAu
+ ( p + p’) grad (div u) =:
3
AijDiDju i.j=l
190
Initial-Boundary Value Problems and the Navier-Stokes Equations
is strongly parabolic since
Here yi E R3 is arbitrary, and yjz’ is the i-th component of yi. Thus the uncoupled u-equation in (6.3.4) is strongly parabolic as long as R(x,t) is bounded. The uncoupled p-equation in (6.3.4) is symmetric hyperbolic since it is of firstorder and scalar. We consider also the inviscid case p = p’ = 0. Except for zero-order terms and forcing functions, one obtains the first-order system (2.4.2) where U , V, W , R are now (known) functions of ( x , t). The symbol P ~ ( xt, ,iw) equals
o u o R O O
and the symmetrizer is
H ( x , t, w) =
v o o
0
o o v O
(i i i i),
R
0
h=tc/R2
Thus the linearized inviscid compressible equations form a strongly hyperbolic system if K = ( d r / d p ) ( R ) > 0. Again, as in the one dimensional case, the symmetrization of the system is equivalent to the introduction of a scaled density.
6.4. Short-Time Existence for Nonlinear Systems Local (in time) existence results for nonlinear strongly hyperbolic, parabolic, and mixed systems in s space dimensions can basically be derived in the same way as in the case of one space dimension. However, the proofs become technically more difficult because the Sobolev inequalities depend on the number s of space dimensions. We first treat nonlinear symmetric hyperbolic systems in
The Cauchy Problem for Systems in Several Dimensions
191
detail. Then corresponding existence results for parabolic and mixed systems are stated.
6.4.1. Nonlinear Symmetric Hyperbolic Systems Consider a symmetric hyperbolic problem
c Y
(6.4.1)
'uf =
Bj(u)Dju, Bj = Bj',
j=l
u(z,O)= fb),f E
(6.4.2)
c-,
f ( ~ ) f(z
+ ej).
As in the one dimensional case, we assume uniform bounds for the (real) coefficients and their derivatives, i.e., 8
C
(6.4.3)
Bj(u)I 5
~
p
p: = 0, 1 , 2 , .. . ,
j=l
for all arguments u E R7L.These global bounds can be replaced by local bounds in a neighborhood of the initial data if one applies the cut-off technique described in Section 5.2. We start with an a priori estimate of 1 1 ~ 1 1 ~ ~ + 2As . it turns out, if this norm is estimated in some time interval, then higher derivatives can be bounded in the same interval. In the proofs we repeatedly use the Sobolev inequality (see Theorem A.3.6) JuI,
S
5 cllullHh if k > -. 2
It implies that
ID'uI,
5 c(IuIJHk if IvI + [s/21+ 1 5 k.
Here [s/2] is the largest integer 5 s/2, and c is a numerical constant independent of u, 2).
Suppose that u solves (6.4.1) in 0 5 t 5 T . Then
Lemma 6.4.1.
where C depends only on the constants KO,.. . ,Ks+2 of (6.4.3). Proof. Differentiation of (6.4.1) yields 9
8
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Initial-Boundary Value Problems and the Navier-Stokes Equations
where
v = (VI,.. . , v3), cr = ( 0 1 , . ..a,), are multi-indices with a
0’ =
( a ; ,. . . , a;,
+ a’ = v and
c, = (VI!...v~9!)/(al! ... as!a ; ! ...a;!).
Therefore, the time derivative
can be estimated in terms of (6.4.4)
(D”U,Bj(U)DjDVU),
j = 1 ,... , s ,
together with (6.4.5)
)
( D ” u , D n ( B j ( u ) ) D B u, 1 5 la1 5 IvI, IcrI
+
= IvI
+ 1.
The symmetry Bj = B; and integration by parts yield (V,Bj(U)DjV)= -
- (Bj(U)DjU,V),
and thus
To estimate a term (6.4.5), we apply the chain rule and write
Here each C, consists of partial derivatives of the coefficient Bj w.r.t u (and C, acts as an r-linear map on D“’ u . . . Dnru.) According to assumption (6.4.3), the C, are bounded. Consider first the case
la1
+ [s/21 + 1 5 s + 2 .
Then (by Sobolev’s inequality) (6.4.6)
IDuauJ, 5 constlluIIHs+2,
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The Cauchy Problem for Systems in Several Dimensions
since
[ail
+ [s/2] + 1 5 s + 2. Therefore,
r=l
Secondly, let
+
Jal [s/21+ 1 thus
> s + 2;
1/31 + [s/21 + 1 I s + 2.
(Observe that la(
+ 1/31 = IvI + 1 5 s + 3.) Then JD('UJ~
5 const
~lu~~~.+z,
and we find that
If each laL[ is
I s + 1 - [s/2], then (6.4.6) applies; hence 1lD"'u.. . D'ru(( 5 const llullh3+2.
If, say, la1I > s + 1 - [s/2], then all other laL/ satisfy Ja,I< s therefore JJD'lu... D't uII
I llD''ulJ
+ 1 - [ s / 2 ] , and
IDu2uIx,.. . IDu7ulx 5 const
IIU~~L.~+~.
Summarizing, the estimate s+2
is shown, and the lemma is proved. The following result is an immediate implication of the previous lemma.
Corollary 6.4.2. There is a time T > 0 depending on 1) f 1IHa+', but not on higher derivatives of f ,with the following property: If u solves (6.4.1). (6.4.2) in 0 I tI T , then Ilu(.,t ) l l H . + 2
5 211f ( l H s + 2 in 0 5 t 5 T .
194
Initial-Boundary Value Problems and the Navier-Stokes Equations
Proof. If f = 0, then u
= 0. Thus let f $ 0, and suppose that y ( t ) solves
where C is determined as in Lemma 6.4.1. Then (by Lemma 4.1.2)
for some T > 0. We shall now show that all higher derivatives of the solution can be estimated in the same time interval 0 I t 5 T . The existence of a smooth solution is assumed. denote the time of Corollary 6.4.2. For Let T = T(JlfllH.+2) Lemma 6.4.3. every p = 0 , 1 , 2 , .. ., there is a constant M p , depending on 11 f J J H ~ +with ~+~,
Proof. For p = 0 the result is shown. Assuming it is proven up to p - 1, we shall prove it for p > 0. To this end, let lvl = s 2 + p . As in the proof of Lemma 6.4.1, we can bound
+
(D”u,Bj(U)DjDVU) 5 c211D”u(12. It remains to estimate
195
The Cauchy Problem for Systems in Several Dimensions
Case 1.
la1
Then
+ [s/2] + 1 5 s + 1 + p . ( D " ( B ~ ( U )5 ) ( const ~ by (6.4.7),
Case 2.
la(
Then
+ [s/2] = s + 1 + p ;
+ [s/21+ 1 I:s + 2 + p.
thus
I CII~IIH*+~+P,
ID"L and it remains to estimate
all Igil I:s + p - [s/2]. The estimate of (6.4.8) follows by (6.4.7). Case 2a.
Case 26.
Then 01 = a ,
10, I > s + p - [s/2l. T
, and thus
=1
+ +
-t- [s/2] > s 1 p ; thus In this case IDfluI, I: const by (6.4.7).
Case 3.
IPI + [s/2l+ 1 5 s + 1 + p .
+
all Ioil I:s p - [s/2]. We have bounded the term (6.4.8) in Case 2a. Case 3a.
lolI > s
Case 36.
We cannot have s
1021
+ p - [s/21.
> s + p - [s/2] since otherwise
+ 2 + p 2 la1 2
Thus IDuiuIw, i
101
I + lo21 2 2s + 2p - 2[s/21 + 2.
# 1 , is bounded by (6.4.7), and llD"'.Ull I
I Il'L1IlH.+2+P.
Il~llHl-l
To summarize, we have shown that d dt
- ( D Y u , D U u )5 const
1(u(L((&J+2+pr Iu(
=s
+2 +p .
196
initial-Boundary Value Problems and the Navier-Stokes Equations
If we use the notation
then
and the induction step can be completed. We shall now sketch the proof for the existence of solutions. To this end, we consider the sequence of functions uk = u k ( z ,t ) defined by the iteration
c V
?$+I
=
Bj(Uk)DjUk+l, U k + l ( Z ,
0) = f (z), u q z , t ) = f (z).
j=l
Note that each function u k is determined by a linear equation. In the same way as in Section 4.1.4, we can show the analogues of Lemma 4.1.5 and Lemma 4.1.6. The techniques to obtain the estimates are the same as used in the proofs of Lemma 6.4.1 and Lemma 6.4.3.
Lemma 6.4.5.
For each j = s
+ 3, s + 4 , . . . there exists Kj with
/Id(., t ) l l ~5 > Kj
in 0 5 t 5 TI,
where TI is determined in the previous lemma. As in Section 4.1.4, we can use the uniform smoothness of the sequence 5 t 5 TI and employ Gronwall's Lemma to show the existence of a C"-solution in 0 5 t 5 Tl. Uniqueness of a solution follows as in the one dimensional case: see Lemma 5.1.1. uk in 0
Theorem 6.4.6. Consider the 1-periodic Cauchy problem (6.4. l), (6.4.2). The problem has a unique C"-solution u = u(x,t ) , dejned for 0 5 t 5 T . The time T > 0 depends on 11 f I ( H a + 2 , but not on higher derivatives o f f .
The Cauchy Problem for Systems in Several Dimensions
197
6.4.2. The Compressible Euler Equations
In two space dimensions the equations without forcing read Ut
(;),+(a Pt
+ U U , + vuy + -1p , P
= 0,
+ up, + upy + p(us + u y ) = 0,
or in matrix form
f br;))
v,
+ uu, + uvy + -P1p ,
= 0,
P = r(p).
(,)..(! ; t..) (;) 0
=o.
0
4!
If r ' ( p ) = $ ( p )
> 0, then we can introduce a new density p by
This transformation leads to a symmetric hyperbolic system. In 3D we can proceed in exactly the same way.
6.4.3. More General Nonlinear Systems The local existence result stated in Theorem 6.4.6 can be generalized without difficulties to
Symmetric hyperbolic systems
c 9
u, =
B3(2,t,21)D3U+F(T,t,21),BJ =I?;.
j=1
where BJ and F depend smoothly on all variables. One can also treat
Strongly parabolic systems
c Y
ut
= E
2,3=I
Dz(A,,(~,t))D3u+F(~,D t , 1I M ~,..., . D,u)
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Initial-Boundary Value Problems and the Navier-Stokes Equations
and obtain an existence interval depending on E > 0. If the system becomes symmetric hyperbolic for c = 0, then an existence interval 0 5 t 5 T, T independent of E > 0, can be established. For E + 0 the solutions of the parabolic problem converge (along with all their derivatives) to the solution of the hyperbolic problem. Proofs of these results have been given in Chapter 5 for one space dimension, but all arguments generalize. Similarly, we obtain local existence for Mixed hyperbolic-parabolic systems
C Y
=
D i ( A i j ( ~t ), Dju)
+ FI(x, t,
U,U,
Du, Du),
i,j=l
1 Bj(x,t,u,u)Djv + F ~ ( xt ,, ~ ,Du), 9
ut =
U,
Bj = Bj*.
j=I
Further generalizations to general parabolic systems, strongly hyperbolic systems (with smooth symmetrizers) and mixed systems can be obtained by changing the Lz-norm. We refer to Sections 6.1.3 and 6.2.2. These results establish short-time existence for the compressible viscous or inviscid Navier-Stokes equations. In the inviscid case, the usual assumption of d r / d p > 0 for the equation of state p = r ( p ) is required to ensure hyperbolicity.
6.5. A Global Existence Theorem in 2D To begin with, we consider a parabolic system where the first-order terms are in so-called self-adjoint form:
(In this section E is arbitrary but fixed.) As before, we assume an initial condition (6.5.2)
u(x,y,O) = f(x,y), f E C",
f 1 - periodic in x and in y.
According to Section 6.4, there is a time T > 0 and a C"-solution u defined for 0 5 t < T. We use the notation
q t ) = IID$L(.,t)112 + IlD;u(., t)112, and start with a simple a priori estimate.
j = 1 , 2 , .. . ,
199
The Cauchy Problem for Systems in Several Dimensions
Lemma 6.5.1. Suppose that IL is a smooth solution dejined for 0 5 t < T . Then there are constant K1, K2 with
lIu(.,t>ll I KI,
(6.5.3)
0 I t < T,
(6.5.4)
Proof. Integration by parts gives us I d 2 2 dt Integrating w.r.t. t, we find that
- -lJu(., t)Jl = ( u ,ut) = t(u, Au) = -cJ:(t).
JO
and the lemma is proved. Now consider a parabolic system (6.5.5)
~t = A(u)uz
+ B(u)uy + EAU,
t
> 0,
where A , B E C" are not necessarily symmetric. We assume that the coefficients grow at most linearly for large arguments: (6.5.6)
IA(u)I
+ IB(u)II K3(1 + lul),
u E R".
The following lemma contains the key result of our global existence theorem.
Lemma 6.5.2. Suppose that u, is a smooth solution of (6.5.5), (6.5.2) defined for 0 5 t < T . Ifwe assume the bounds (6.5.3), (6.5.4), (6.5.6), then
are finite.
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Initial-Boundary Value Problems and the Navier-Stokes Equations
One shows easily that
and with assumption (6.5.4) we conclude that 1
I Jl(O)exp{ 5C3(T + K d } . Thus we have proved a bound for J l ( t ) .
201
The Cauchy Problem for Systems in Several Dimensions
If we use (6.5.7) again and observe that J;
lT
I l)LIt~11~, then
J i ( r ) d r < 00
follows. The special Sobolev inequality implies that lemma is proved.
so lul& d r < T
00,and
the
Let us assume that also the first derivatives of the coefficients in (6.5.5) grow at most linearly for large arguments:
(6.5.8)
lAu(u)l + l&(u)l
I K4(1 + lu0,
u E R".
Then we can apply the same technique to the differentiated differential equation (6.5.5). As a result, one obtains
Theorem 6.5.3. Suppose that u E C" is a solution of (6.5.5), (6.5.2)defined for 0 5 t < T . We assume the bounds (6.5.3), (6.5.4) for u = u(x,t ) and the bounds (6.5.6), (6.5.8)for the coefficients of the differential equation. Then
are finite, and consequently u can be extended as a smooth solution beyond T . If a priori bounds (6.5.3), (6.5.4) (with Kj = h;(T))hold for any finite time interval, then u exists for all time.
202
Initial-Boundary Value Problems and the Navier-Stokes Equations
The inequality for ( d / d t ) J i yields
and the desired bound follows from Lemma 6.5.2. This process can be continued, and by induction we obtain bounds for all derivatives. The remaining statements can be shown by the general arguments given at the beginning of Section 5.4.
Notes on Chapter 6 Petrovskii (1938) proved the existence of solutions of the Cauchy problem for strictly hyperbolic equations. The proof was simplified by k r a y (1953), who constructed a symmetrizer. Kreiss (1963) generalized the construction to the case where the algebraic multiplicity of the eigenvalues of the symbol P(z,t , iw) does not change. In this case the symmetrizer can be chosen as a quotient of differential operators. The smoothness of the symmetrizer is a major problem when the multiplicity of the eigenvalues changes. Besides the trivial case when the system is symmetric and H 3 I - no general theorem is known. Recently Clarke Hernquist (1988) was able to construct a smooth symmetrizer for a particular class of symbols with changing multiplicity. The class of corresponding differential equations was introduced by John (1978). For the construction of the symmetrizer for parabolic systems see also Mizohata (1956) and Kreiss (1963).
The Cauchy Problem for Systems in Several Dimensions
The results of Chapter 3 about linear parabolic, strongly hyperbolic, and mixed systems can mostly be generalized from one space dimension to an arbitrary number of space dimensions. These generalizations are carried out in the first three sections of this chapter. In particular, we show well-posedness of the Cauchy problem for the linearized compressible Navier-Stokes and Euler equations. Section 6.4 treats short-time existence for nonlinear systems; we carry out the details for the symmetric hyperbolic case and sketch generalizations to systems of different type. For a special class of nonlinear parabolic systems in two space dimensions, we will prove all-time existence in Section 6.5.
6.1. Linear Parabolic Systems We consider second-order parabolic systems of the general form (6.1.1)
ut =
2
I.J=i
Dt(A,,D,u)
+
2
B,D,ir
+ Cu + F.
,=I
Here s is the number of space dimensions and D , is the operator D, = i)/ds, . The coefficient matrices A,, = A,,(z,t), €3, = B , ( z , t ) , C = C ( s . t )and the forcing function F = F ( z ,t ) are assumed to be C”-smooth, real, and 1-periodic
I77
178
Initial-Boundary Value Problems and the Navier-Stokes Equations
in each component xi,i = 1 , . .. , s. The concept of parabolicity will be defined below. For the function u(x,t) an initial condition (6.1.2)
u(z:O)= f(x), f E C",
f I-periodic in each xi
is given, where f takes values in R". As before, we seek a smooth solution u = u(x, t) taking values in R" which is 1-periodic in each xi, i = 1 , . . . , s. In Section 6.1.1 the concept of strong parabolicity will be defined; it leads to a simple derivation of solution-estimates. Existence of a solution can again be shown by proving the analogous estimates for a difference scheme. This will be sketched in Section 6.1.2. Existence, uniqueness, and estimates can be generalized from strongly parabolic systems to systems for which a pointwise definition of parabolicity is used. For our applications this generalization is of minor importance, however, and we only sketch the proof.
6.1.1. Estimates for Strongly Parabolic Systems The following definition generalizes our previous concept of strong parabolicity to systems in any number of space dimensions.
Dejnition. The system (6.1.1) is called strongly parabolic in 0 5 t 5 T if there exists a constant 6 > 0 such that the estimate
i= I
holds for all x E R", 0 5 t 5 T, and all vectors yi E R", i = 1 , . . . , s. The basic result about existence, uniqueness, and estimates is
Theorem 6.1.1. If(6.1.1) is stronglyparabolic in 0 5 t 5 T , then the Cauchy problem (6.1. I ) , (6.1.2) has a unique classical solution u = u(x,t ) in 0 5 t 5 T . The solution u is a Cm-function. For each p = 0 , 1,2,. . ., there is a constant K p with
The constant K p depends on 6 and the maximum norm of the coefficients of (6.1.1) and their derivatives of order 5 p , but not on F, f.
The Cauchy Problem for Systems in Several Dimensions
179
Proof. We start with the estimates. Integration by parts yields I d 2 dt
- - ( u , u ) = ( u ,u t ) S
S
Therefore, (6.1.3) follows for p = 0 if we apply Lemma 3.1.1. This estimate also shows the uniqueness of a classical solution. To estimate first derivatives, we differentiate the equation (6.1.1) with respect to z,, i = 1 , .. , , s , and obtain a system of similar type for the vector
(DIU,D2u,. .. . D,u). The coefficients of this system depend on A , j , B,, C and the first derivatives of these matrices. The desired estimate follows as above. Estimates for higher derivatives are obtained by repeated differentiation.
In the next section we sketch the proof of the existence of
u.
6.1.2. Existence via Difference Approximations Let h = 1/N denote a gridlength, N a natural number. For any multi-index
v = (v1... . ,vs), vj E z. let z, = (hvl , . . . , h ~ , denote ~ ) the corresponding meshpoint. We consider gridfunctions v, = v(z,,) which are assumed to be 1-periodic in each coordinate direction. By EJ we denote the translation operator in the j-th coordinate:
Ejuv = Ejv(z,)= u ( z ,
+ he,),
e3 = (0.... , 0 , 1 , 0 , . . . , 0) with 1 at position j The powers of EJ are
EJPu,, = v(s,
+ phe,),
p E Z,
and EJ”= I is the identity. The forward, backward, and centered difference operators in the 3-th coordinate direction are
180
Initial-Boundary Value Problems and the Navier-Stokes Equations
1 2
DOj = - (D+j
+ D-j),
j = 1 , . .., s .
Using these notations, we approximate (6.1.1) by the system of ordinary differential equations
i=l
If we introduce a discrete scalar product and norm for gridfunctions by
then all computational rules of Section 3.2.2 carry over in an obvious way to the multi-dimensional case. Therefore, we can estimate the solutions of (6.1.4) and their divided differences independently of h. We just mimic the continuous process. As in the one dimensional case, we can interpolate v,(t) = v,h(t)with respect to the z-variables by Fourier polynomials w"(z, t). (See Appendix 2.) These are uniformly (with respect to h) smooth; in the limit h + 0 we obtain a solution u = u ( z ,t ) of the Cauchy problem. There is no essential difference from the one dimensional case. 6.1.3.
General Linear Second-Order Parabolic Systems
In Section 2.4.3 we have defined parabolicity for problems with constant coefficients. Applying this concept to all frozen-coefficient problems, we are lead to: Definition. The system (6.1.1) is called parabolic in 0 5 t 5 T ifthere is a constant 6 > 0 such that for all x E R",0 5 t 5 T , and all w = (wI, . . . ,w,)E Rs the eigenvalues K of the matrix I
satisfy Re
K
2 61wI2.
The Cauchy Problem for Systems in Several Dimensions
181
One can prove
Theorem 6.1.2. The statements of Theorem 6.1.1 are valid if the assumption of strong parabolicity is replaced by the assumption of parabolicity. We only sketch the proof of this result. The essential point is to obtain the a priori estimates (6.1.3); then existence of a solution follows as before by use of a difference scheme. To obtain the estimate (6.1.3) for p = 0, first fix a point (so, t o ) and consider the constant-coefficient parabolic operator s
~ 2 ( 5 n t, o , D )
=
C 1
D , ALJ(so, t o ) D,.
]=I
If we proceed as in Section 2.7 - but replace Fourier transforms by Fourier expansions - we can construct an inner product (u,
to) =
C(a(u), ~ ( z oto, , ~)s(w)) W
with
(. P2(zo. to. D)u)H(c",t") + (p2(zo,to. O u , . ) H ( s o , t r , ) <- -6
c 5
llDL4*
?=I
for all smooth functions u = u(z). Using the corresponding norms and a partition of unity argument as in Section 3.2.5, we obtain the basic energy estimate as before. Estimates for derivatives can be obtained by differentiating the differential equation. As noted in Section 3.2.6, a basic property of parabolic differential operators is their smoothing ability. This smoothing property is also valid for equations in more than one space dimension and can be proved as in the one dimensional case
6.2. Linear Hyperbolic Systems A first-order system
(6.2. la)
ut
= P ( z ,t , d/dz)u
+ C(a,t)u + F ( z .t )
is called symmetric hyperbolic if P ( z . t . d / d z ) has the form (6.2. I b )
c{
1 "
P ( z ,t , d / d x ) = ~2 ,=I
B 3 ( r ,t)D,u
+ D3(BJ(x, t)~)},
182
Initial-Boundary Value Problems and the Navier-Stokes Equations
and B j ( x ,t ) = Bj+(z,t ) for all x , t. As before, the coefficients B j , C , and the forcing function F are assumed to be real, Cm-smooth, and 1-periodic in each Xj.
To obtain existence for the Cauchy problem (6.2. l), (6.1.2), we consider first the parabolic problems ut = t AU + P ( x , t , d / d x ) u + CU+ F,
t
> 0.
The existence result of Theorem 6.1.1 applies, and the estimates are independent of E > 0. For E -+ 0, one obtains a solution of the symmetric hyperbolic problem. A basic property of hyperbolic equations is the “finite speed of propagation”. We formulate the result in the following theorem; the proof uses the convergence of an explicit difference scheme.
Theorem 6.2.1.
Define
The solution-value u(x0,to) depends only on those values of F ( x ,t ) for lying in the cone
1x0 - 21 and only on those values off
I a(to - t ) ,
( 2 )for
05t
( 2 ,t )
I to,
x lying in the interval 1x0 - x(5 ato.
h I
FIGURE 6.2.1. Domain of dependence and mesh.
X
183
The Cauchy Problem for Systems in Several Dimensions
Proof. 1. As in Section 6.2.1, we introduce meshpoints x u , v E Z". but discretize in time also. Let k > 0 denote the time step-size. We approximate (6.2.1) by the difference approximation
( I - k C ( z u ,t ) ) li,(t
+ k ) = ( I + kC(x,, t ) ) v u ( t - k ) + 2kQ(xu, t ) v,At) + 2 k F ( x u ,t ) ,
(6.2.2)
t
= k . 2k. 3 k , . . . ,
Q(xu,t ) t ~ u ( t = )
" 51 C { B j ( r , , t)Doju,(t) + Do] ( B j ( z u ,t)v,(t))} . )=I
Here 1 Doj = -(D+, 2
+ D - j ) = -2h1( E j
I
- Ej- )
denotes the usual centered difference operator. The three level difference scheme (6.2.2) is supplemented by the starting conditions
(6.2.3) ~ ~ (=0f ( )x u ) , v u ( k )= f ( z u ) k { Q(zu.0)
+
+ C(s,. O)}f(x,) + k F ( x U 0). .
+
We fix a relation k = h / ( a q ) , 77 > 0 arbitrary, and send h + 0. The difference equations (6.2.2), (6.2.3) define an approximation t i = uh for the exact solution u, and we will prove the convergence
The domain of dependence of uh belongs to the cone 1x0 - 21
I ( a + q)(to - t ) , 0 I t
I to.
Since q > 0 is arbitrary and u is smooth, the assertion of the theorem follows. 2. It remains to show (6.2.4). First note that the operators Doi are antisymmetric:
184
Initial-Boundary Value Problems and the Navier-Stokes Equations
We give a simple bound for Q ( t ) : The estimate
implies that
L
j=l
The smooth solution u = u ( z ,t ) of the differential problem satisfies
Thus we can consider u,(t) as a solution of perturbed difference equations:
Here $,(t), 9, are bounded independently of h and k . The difference solves
Taking the inner product with m,(t find that
+ k ) + w,(t
-
UJ= u--v
k ) and summing over v, we
The Cauchy Problem for Systems in Several Dimensions
185
This implies a lower bound for L(t),
and therefore the above recursive estimate gives us that
+
U t + k ) I L ( t ) kc&t
+ k ) + kc,L(t) + k(k2 + h 2 ) 2 C 4 .
Hence the auxiliary quantities L ( t ) also satisfy
L(t + k ) 5 ( I
+ k q ) L ( t )+ k(k2 + h 2 ) 2 C 6 .
Initially, at t = k , we have L ( k ) = llw(k)11; = O(k4), and by a discrete analog of Gronwall's Lemma (Lemma 3.1.1) it follows that
L ( t ) = O((k2 + h 2 ) 2 ) , 0 5 t 5 to. Since L(t) bounds
Ilci~(t)ll;~ , the
convergence (6.2.4) is proved.
186
Initial-Boundary Value Problems and the Navier-Stokes Equations
The existence result mentioned above for symmetric hyperbolic systems can be extended to more general hyperbolic equations. Consider the first-order system (6.2.la) where the operator P ( z . t,a/az)has the form (6.2.1b), but the matrices B,(z, t) are not necessarily symmetric. We call s
P(z,t,iw) =
iC
BJ(z,t)wj. w E R", IwI = 1.
J=l
the symbol of the differential operator P and make the The system (6.2.1) is called strongly hyperbolic if there exists a smooth 1-periodic symmetrizer; i.e., there is a 1-periodic positive definite Hermitian matrix function
Definition.
H ( x , t , w ) , z E R". t 2 0, w E R", lwl = 1 , depending smoothly on all variables, with
+
H ( r . t , w ) P ( z ,t , iw) P*(Z,t , i w ) H ( z .t , w)= 0. This definition generalizes the conditions discussed in Section 2.4.1 for the multi-dimensional constant-coefficient case and the conditions in Section 3.3.1 for the one dimensional variable-coefficient case, but it requires in addition smooth dependence on (2, t. w). (In the constant-coefficientcase smooth dependence on w was not essential.) If we require that all frozen-coefficient problems obtained from (6.2.1) be strongly hyperbolic in the sense of Section 2.4.1, then we can use the matrices
S ( s . t . w ) , w E R". IwI = 1 . of Theorem 2.4.1 and obtain the symmetrizer H = S'S; compare Lemma 2.4.2. The only extra assumption in the above definition is the requirement of periodicity and smoothness of H = H ( z . t , w ) . The matrix S - ' ( x , t , w ) contains in its columns the eigenvectors of P ( z .t , iw). Thus, if the eigenvalues of P ( r ,t. iw)are all purely imaginary and if linearly independent eigenvectors can be chosen as smooth periodic functions of ( z , t . w ) , then the system is strongly hyperbolic. This is the case if the system is strictly hyperbolic, i.e., the eigenvalues of P ( z ,t , iw)are purely imaginary and always distinct. One can prove
Theorem 6.2.2. If the system (6.2.1) is strongly hyperbolic, then the Cauchy problem is well-posed. For every set of smooth data F = F ( z . t ) , f = f(x)
The Cauchy Problem for Systems in Several Dimensions
187
there exists a unique smooth solution. Estimates as in the one dimensional case are valid; i.e., the solution is as smooth as the data. Sketch of the proof. We use the theory of pseudodifferential operators and refer to Eskin (1981, Chapter 5) or Nirenberg (1973). To conform with the usual framework, we do not assume that the data and coefficients are 1-periodic. Instead, we assume that F, f E C" and that the coefficients are constant outside a bounded set. Because of the finite speed of propagation, this is no restriction. Let 4j = 4j(T) denote a partition of unity where the support of 4j has diameter dJ and sup, d3 is sufficiently small. Also, at every point T at most p functions #3 are different from zero, p independent of T . We define the pseudodifferential operator H(t) by
For sufficiently small d j
defines a scalar product which is equivalent to the usual Lz-scalar product. To prove this, we define
H,(t)v(c)= and note that
L.
H ( z , , t , W / I C Ja ( ( w) ) dP w , (xJ ~E. ~ ) 4,, support
This ensures that
can be estimated from below by const 1 1 ~ 1 1 . Let 4 = $(lwl),# E C", denote a "cut-off' function with
We write (at each t )
H=H+HR?
Initial-Boundary Value Problems and the Navier-Stokes Equations
188
where
6(w)dw. With these notations,
+ Re ( u j ,HPuj) = (ujl (HP + P*H*)uj + Re ( q HPuj). , 2
Re ( u j ,HPuj) = Re (uj ,HPuj)
Again, HP + P*H*is a pseudodifferential operator. The calculus of pseudodifferential operators implies that its symbol has the form
4(lwl) (m? w/lwl)P(z,t , iw)+ P*(X,t , i w ) H ( z ,t ,w/lwl)) t l
+ S(Z, t , w).
+
The part multiplied by 4 vanishes, and S(z, t , w) is bounded. Hence, HP P*H*is a bounded operator. Because the factor 1 - 4(lwl) in the definition of H cuts off high frequencies, the operator HP *isalso bounded. Therefore one obtains that d (u, dt u ) H ( t ) 5 const ( u !U ) H ( t )
+
(u1
F)H(t):
and the basic energy estimate follows.
6.3. Mixed Hyperbolic-Parabolic Systems and the Linearized Navier-Stokes Equations There are no problems in obtaining existence and uniqueness results, and estimates of smooth solutions for mixed hyperbolic-parabolic systems of the form
~t = P I ( z t, , d
+
+
/ d ~ ) v R ~ [ (tX , d, / d ~ )G~I .
Here we assume that the uncoupled systems (6.3.2)
+ G, Vt = P~(x, t ,d / d z ) T + F
Tit = P ~ ( x t ,,d / d ~ ) i i
are second-order strongly parabolic and first-order symmetric hyperbolic, respectively. The coupling terms R12 and R ~ are I general first-order operators.
189
The Cauchy Problem for Systems in Several Dimensions
The proof proceeds as in the one dimensional case; we refer to Section 3.4.1. The solutions are as smooth as the data. One can also generalize this result to the case where the uncoupled systems (6.3.2) are second-order parabolic and first-order strongly hyperbolic, respectively. The proof for this more general case requires that one modify the Lznorm to obtain the estimates. We shall now discuss how these results apply to the linearized Navier-Stokes equations. The (nonlinear) viscous compressible system reads p
(6.3.3)
D -u Dt
+ grad p = p A u + ( p + p ’ ) grad (div u ) + pF,
D Dt
+ p div u = 0,
-p
p = r(p),
D - d d d d - - + u - + 2) - + w -. Dt dt dy dz dx
-
Suppose U(x, t ) . P(x,t ) , R(x,t ) are smooth functions with R > 0 and P = r(R); these functions are considered as an exact or approximate solution of (6.3.3). We substitute
u=U+U’,
p=P+p’,
p=R+p’
into (6.3.3) and neglect all terms which are quadratic in the corrections u’, p’. p’. If we drop the ’ sign in our notation, we obtain the linear equations
D RTu Dt
D
(6.3.4)
-
Dt
D + p Dt U = pAu + ( p + p’) grad (div u) + G I
+
K
grad p
D
Dt
p
grad)U
d = at-a + u -+v -d + w K
-
+ R(u.
7
d dz’
-
dr
= -(R),
dP
+ R div u + u . grad R + p div U = G2.
The inhomogeneous terms G I , G2 are determined by U , P, R and vanish if U, P, R solve (6.3.3) exactly. Clearly, the equations (6.3.4) are of the general form (6.3.1) with u = u, 2) = p . For p > 0, p’ 2 0, the second-order operator
pAu
+ ( p + p’) grad (div u) =:
3
AijDiDju i.j=l
190
Initial-Boundary Value Problems and the Navier-Stokes Equations
is strongly parabolic since
Here yi E R3 is arbitrary, and yjz’ is the i-th component of yi. Thus the uncoupled u-equation in (6.3.4) is strongly parabolic as long as R(x,t) is bounded. The uncoupled p-equation in (6.3.4) is symmetric hyperbolic since it is of firstorder and scalar. We consider also the inviscid case p = p’ = 0. Except for zero-order terms and forcing functions, one obtains the first-order system (2.4.2) where U , V, W , R are now (known) functions of ( x , t). The symbol P ~ ( xt, ,iw) equals
o u o R O O
and the symmetrizer is
H ( x , t, w) =
v o o
0
o o v O
(i i i i),
R
0
h=tc/R2
Thus the linearized inviscid compressible equations form a strongly hyperbolic system if K = ( d r / d p ) ( R ) > 0. Again, as in the one dimensional case, the symmetrization of the system is equivalent to the introduction of a scaled density.
6.4. Short-Time Existence for Nonlinear Systems Local (in time) existence results for nonlinear strongly hyperbolic, parabolic, and mixed systems in s space dimensions can basically be derived in the same way as in the case of one space dimension. However, the proofs become technically more difficult because the Sobolev inequalities depend on the number s of space dimensions. We first treat nonlinear symmetric hyperbolic systems in
The Cauchy Problem for Systems in Several Dimensions
191
detail. Then corresponding existence results for parabolic and mixed systems are stated.
6.4.1. Nonlinear Symmetric Hyperbolic Systems Consider a symmetric hyperbolic problem
c Y
(6.4.1)
'uf =
Bj(u)Dju, Bj = Bj',
j=l
u(z,O)= fb),f E
(6.4.2)
c-,
f ( ~ ) f(z
+ ej).
As in the one dimensional case, we assume uniform bounds for the (real) coefficients and their derivatives, i.e., 8
C
(6.4.3)
Bj(u)I 5
~
p
p: = 0, 1 , 2 , .. . ,
j=l
for all arguments u E R7L.These global bounds can be replaced by local bounds in a neighborhood of the initial data if one applies the cut-off technique described in Section 5.2. We start with an a priori estimate of 1 1 ~ 1 1 ~ ~ + 2As . it turns out, if this norm is estimated in some time interval, then higher derivatives can be bounded in the same interval. In the proofs we repeatedly use the Sobolev inequality (see Theorem A.3.6) JuI,
S
5 cllullHh if k > -. 2
It implies that
ID'uI,
5 c(IuIJHk if IvI + [s/21+ 1 5 k.
Here [s/2] is the largest integer 5 s/2, and c is a numerical constant independent of u, 2).
Suppose that u solves (6.4.1) in 0 5 t 5 T . Then
Lemma 6.4.1.
where C depends only on the constants KO,.. . ,Ks+2 of (6.4.3). Proof. Differentiation of (6.4.1) yields 9
8
192
Initial-Boundary Value Problems and the Navier-Stokes Equations
where
v = (VI,.. . , v3), cr = ( 0 1 , . ..a,), are multi-indices with a
0’ =
( a ; ,. . . , a;,
+ a’ = v and
c, = (VI!...v~9!)/(al! ... as!a ; ! ...a;!).
Therefore, the time derivative
can be estimated in terms of (6.4.4)
(D”U,Bj(U)DjDVU),
j = 1 ,... , s ,
together with (6.4.5)
)
( D ” u , D n ( B j ( u ) ) D B u, 1 5 la1 5 IvI, IcrI
+
= IvI
+ 1.
The symmetry Bj = B; and integration by parts yield (V,Bj(U)DjV)= -
- (Bj(U)DjU,V),
and thus
To estimate a term (6.4.5), we apply the chain rule and write
Here each C, consists of partial derivatives of the coefficient Bj w.r.t u (and C, acts as an r-linear map on D“’ u . . . Dnru.) According to assumption (6.4.3), the C, are bounded. Consider first the case
la1
+ [s/21 + 1 5 s + 2 .
Then (by Sobolev’s inequality) (6.4.6)
IDuauJ, 5 constlluIIHs+2,
193
The Cauchy Problem for Systems in Several Dimensions
since
[ail
+ [s/2] + 1 5 s + 2. Therefore,
r=l
Secondly, let
+
Jal [s/21+ 1 thus
> s + 2;
1/31 + [s/21 + 1 I s + 2.
(Observe that la(
+ 1/31 = IvI + 1 5 s + 3.) Then JD('UJ~
5 const
~lu~~~.+z,
and we find that
If each laL[ is
I s + 1 - [s/2], then (6.4.6) applies; hence 1lD"'u.. . D'ru(( 5 const llullh3+2.
If, say, la1I > s + 1 - [s/2], then all other laL/ satisfy Ja,I< s therefore JJD'lu... D't uII
I llD''ulJ
+ 1 - [ s / 2 ] , and
IDu2uIx,.. . IDu7ulx 5 const
IIU~~L.~+~.
Summarizing, the estimate s+2
is shown, and the lemma is proved. The following result is an immediate implication of the previous lemma.
Corollary 6.4.2. There is a time T > 0 depending on 1) f 1IHa+', but not on higher derivatives of f ,with the following property: If u solves (6.4.1). (6.4.2) in 0 I tI T , then Ilu(.,t ) l l H . + 2
5 211f ( l H s + 2 in 0 5 t 5 T .
194
Initial-Boundary Value Problems and the Navier-Stokes Equations
Proof. If f = 0, then u
= 0. Thus let f $ 0, and suppose that y ( t ) solves
where C is determined as in Lemma 6.4.1. Then (by Lemma 4.1.2)
for some T > 0. We shall now show that all higher derivatives of the solution can be estimated in the same time interval 0 I t 5 T . The existence of a smooth solution is assumed. denote the time of Corollary 6.4.2. For Let T = T(JlfllH.+2) Lemma 6.4.3. every p = 0 , 1 , 2 , .. ., there is a constant M p , depending on 11 f J J H ~ +with ~+~,
Proof. For p = 0 the result is shown. Assuming it is proven up to p - 1, we shall prove it for p > 0. To this end, let lvl = s 2 + p . As in the proof of Lemma 6.4.1, we can bound
+
(D”u,Bj(U)DjDVU) 5 c211D”u(12. It remains to estimate
195
The Cauchy Problem for Systems in Several Dimensions
Case 1.
la1
Then
+ [s/2] + 1 5 s + 1 + p . ( D " ( B ~ ( U )5 ) ( const ~ by (6.4.7),
Case 2.
la(
Then
+ [s/2] = s + 1 + p ;
+ [s/21+ 1 I:s + 2 + p.
thus
I CII~IIH*+~+P,
ID"L and it remains to estimate
all Igil I:s + p - [s/2]. The estimate of (6.4.8) follows by (6.4.7). Case 2a.
Case 26.
Then 01 = a ,
10, I > s + p - [s/2l. T
, and thus
=1
+ +
-t- [s/2] > s 1 p ; thus In this case IDfluI, I: const by (6.4.7).
Case 3.
IPI + [s/2l+ 1 5 s + 1 + p .
+
all Ioil I:s p - [s/2]. We have bounded the term (6.4.8) in Case 2a. Case 3a.
lolI > s
Case 36.
We cannot have s
1021
+ p - [s/21.
> s + p - [s/2] since otherwise
+ 2 + p 2 la1 2
Thus IDuiuIw, i
101
I + lo21 2 2s + 2p - 2[s/21 + 2.
# 1 , is bounded by (6.4.7), and llD"'.Ull I
I Il'L1IlH.+2+P.
Il~llHl-l
To summarize, we have shown that d dt
- ( D Y u , D U u )5 const
1(u(L((&J+2+pr Iu(
=s
+2 +p .
196
initial-Boundary Value Problems and the Navier-Stokes Equations
If we use the notation
then
and the induction step can be completed. We shall now sketch the proof for the existence of solutions. To this end, we consider the sequence of functions uk = u k ( z ,t ) defined by the iteration
c V
?$+I
=
Bj(Uk)DjUk+l, U k + l ( Z ,
0) = f (z), u q z , t ) = f (z).
j=l
Note that each function u k is determined by a linear equation. In the same way as in Section 4.1.4, we can show the analogues of Lemma 4.1.5 and Lemma 4.1.6. The techniques to obtain the estimates are the same as used in the proofs of Lemma 6.4.1 and Lemma 6.4.3.
Lemma 6.4.5.
For each j = s
+ 3, s + 4 , . . . there exists Kj with
/Id(., t ) l l ~5 > Kj
in 0 5 t 5 TI,
where TI is determined in the previous lemma. As in Section 4.1.4, we can use the uniform smoothness of the sequence 5 t 5 TI and employ Gronwall's Lemma to show the existence of a C"-solution in 0 5 t 5 Tl. Uniqueness of a solution follows as in the one dimensional case: see Lemma 5.1.1. uk in 0
Theorem 6.4.6. Consider the 1-periodic Cauchy problem (6.4. l), (6.4.2). The problem has a unique C"-solution u = u(x,t ) , dejned for 0 5 t 5 T . The time T > 0 depends on 11 f I ( H a + 2 , but not on higher derivatives o f f .
The Cauchy Problem for Systems in Several Dimensions
197
6.4.2. The Compressible Euler Equations
In two space dimensions the equations without forcing read Ut
(;),+(a Pt
+ U U , + vuy + -1p , P
= 0,
+ up, + upy + p(us + u y ) = 0,
or in matrix form
f br;))
v,
+ uu, + uvy + -P1p ,
= 0,
P = r(p).
(,)..(! ; t..) (;) 0
=o.
0
4!
If r ' ( p ) = $ ( p )
> 0, then we can introduce a new density p by
This transformation leads to a symmetric hyperbolic system. In 3D we can proceed in exactly the same way.
6.4.3. More General Nonlinear Systems The local existence result stated in Theorem 6.4.6 can be generalized without difficulties to
Symmetric hyperbolic systems
c 9
u, =
B3(2,t,21)D3U+F(T,t,21),BJ =I?;.
j=1
where BJ and F depend smoothly on all variables. One can also treat
Strongly parabolic systems
c Y
ut
= E
2,3=I
Dz(A,,(~,t))D3u+F(~,D t , 1I M ~,..., . D,u)
198
Initial-Boundary Value Problems and the Navier-Stokes Equations
and obtain an existence interval depending on E > 0. If the system becomes symmetric hyperbolic for c = 0, then an existence interval 0 5 t 5 T, T independent of E > 0, can be established. For E + 0 the solutions of the parabolic problem converge (along with all their derivatives) to the solution of the hyperbolic problem. Proofs of these results have been given in Chapter 5 for one space dimension, but all arguments generalize. Similarly, we obtain local existence for Mixed hyperbolic-parabolic systems
C Y
=
D i ( A i j ( ~t ), Dju)
+ FI(x, t,
U,U,
Du, Du),
i,j=l
1 Bj(x,t,u,u)Djv + F ~ ( xt ,, ~ ,Du), 9
ut =
U,
Bj = Bj*.
j=I
Further generalizations to general parabolic systems, strongly hyperbolic systems (with smooth symmetrizers) and mixed systems can be obtained by changing the Lz-norm. We refer to Sections 6.1.3 and 6.2.2. These results establish short-time existence for the compressible viscous or inviscid Navier-Stokes equations. In the inviscid case, the usual assumption of d r / d p > 0 for the equation of state p = r ( p ) is required to ensure hyperbolicity.
6.5. A Global Existence Theorem in 2D To begin with, we consider a parabolic system where the first-order terms are in so-called self-adjoint form:
(In this section E is arbitrary but fixed.) As before, we assume an initial condition (6.5.2)
u(x,y,O) = f(x,y), f E C",
f 1 - periodic in x and in y.
According to Section 6.4, there is a time T > 0 and a C"-solution u defined for 0 5 t < T. We use the notation
q t ) = IID$L(.,t)112 + IlD;u(., t)112, and start with a simple a priori estimate.
j = 1 , 2 , .. . ,
199
The Cauchy Problem for Systems in Several Dimensions
Lemma 6.5.1. Suppose that IL is a smooth solution dejined for 0 5 t < T . Then there are constant K1, K2 with
lIu(.,t>ll I KI,
(6.5.3)
0 I t < T,
(6.5.4)
Proof. Integration by parts gives us I d 2 2 dt Integrating w.r.t. t, we find that
- -lJu(., t)Jl = ( u ,ut) = t(u, Au) = -cJ:(t).
JO
and the lemma is proved. Now consider a parabolic system (6.5.5)
~t = A(u)uz
+ B(u)uy + EAU,
t
> 0,
where A , B E C" are not necessarily symmetric. We assume that the coefficients grow at most linearly for large arguments: (6.5.6)
IA(u)I
+ IB(u)II K3(1 + lul),
u E R".
The following lemma contains the key result of our global existence theorem.
Lemma 6.5.2. Suppose that u, is a smooth solution of (6.5.5), (6.5.2) defined for 0 5 t < T . Ifwe assume the bounds (6.5.3), (6.5.4), (6.5.6), then
are finite.
200
Initial-Boundary Value Problems and the Navier-Stokes Equations
One shows easily that
and with assumption (6.5.4) we conclude that 1
I Jl(O)exp{ 5C3(T + K d } . Thus we have proved a bound for J l ( t ) .
201
The Cauchy Problem for Systems in Several Dimensions
If we use (6.5.7) again and observe that J;
lT
I l)LIt~11~, then
J i ( r ) d r < 00
follows. The special Sobolev inequality implies that lemma is proved.
so lul& d r < T
00,and
the
Let us assume that also the first derivatives of the coefficients in (6.5.5) grow at most linearly for large arguments:
(6.5.8)
lAu(u)l + l&(u)l
I K4(1 + lu0,
u E R".
Then we can apply the same technique to the differentiated differential equation (6.5.5). As a result, one obtains
Theorem 6.5.3. Suppose that u E C" is a solution of (6.5.5), (6.5.2)defined for 0 5 t < T . We assume the bounds (6.5.3), (6.5.4) for u = u(x,t ) and the bounds (6.5.6), (6.5.8)for the coefficients of the differential equation. Then
are finite, and consequently u can be extended as a smooth solution beyond T . If a priori bounds (6.5.3), (6.5.4) (with Kj = h;(T))hold for any finite time interval, then u exists for all time.
202
Initial-Boundary Value Problems and the Navier-Stokes Equations
The inequality for ( d / d t ) J i yields
and the desired bound follows from Lemma 6.5.2. This process can be continued, and by induction we obtain bounds for all derivatives. The remaining statements can be shown by the general arguments given at the beginning of Section 5.4.
Notes on Chapter 6 Petrovskii (1938) proved the existence of solutions of the Cauchy problem for strictly hyperbolic equations. The proof was simplified by k r a y (1953), who constructed a symmetrizer. Kreiss (1963) generalized the construction to the case where the algebraic multiplicity of the eigenvalues of the symbol P(z,t , iw) does not change. In this case the symmetrizer can be chosen as a quotient of differential operators. The smoothness of the symmetrizer is a major problem when the multiplicity of the eigenvalues changes. Besides the trivial case when the system is symmetric and H 3 I - no general theorem is known. Recently Clarke Hernquist (1988) was able to construct a smooth symmetrizer for a particular class of symbols with changing multiplicity. The class of corresponding differential equations was introduced by John (1978). For the construction of the symmetrizer for parabolic systems see also Mizohata (1956) and Kreiss (1963).