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Uniform decay rates for parabolic conservation laws
UNIFORIM
DECAY
RATES
FOR PARABOLIC MARIA
Department
of Mathematics.
Duke
(Received in reoised form 17 ,Vouember Key words and phrases: Parabolic
conservation
E.
CONSERVATION
LAWS
SCHONBEK
University.
Durham,
NC 27706.
U.S.A.
1983; received for publication 21 October 1985) laws, decay
rates.
1. INTRODUCTION
paper we establish optimal uniform rates of decay for solutions of the Cauchy problem with large data to scalar parabolic conservation laws in several space dimensions, IN THIS
where the II,, are constants, and to parabolic systems of conservation
laws
$ + divf(L1) = Au,
(1.2)
where u = U(X, t) E RP, x E R” and f, = (f’, . . . ,fp) with f: RF- R”. Equations of this type arise in continuum mechanics where they serve as models for elastic solids and ideal gases. In an earlier paper [j], we have shown that solutions of the scalar equation (1.1) decay in L’ and L’ with a rate of ten”. It is not difficult to verify that t -n:’ is the optimal rate in L’. On the other hand, one can expect the faster rate of ten,’ m ’ L” based on an analysis of the heat equation. In Section 2, we shall show that solutions of the scalar equation (1.1) decay at the optimal rates t -n “‘-‘;P) in LP for 1 < p < x and at the optimal rate of t-“/I in L”. In order to establish the decay rate we derive an entropy inequality for the function up. Applying Fourier analysis yields an inequality for the Fourier transform of an appropriate power depending on p. A subdivision of the frequency domain into two time dependent subspaces leads to the desired decay rate. In Sections 3 and 4 we show that periodic solutions of system (1.2) decay exponentially in H” and L” for a class of general flux functionsfi With regard to background we recall that decay without a rate of solutions of (1.2) in one spatial dimension was established in [5]. In the present paper we first obtain the exponential decay in the L?-norm. We derive an entropy inequality which expresses the dissipative mechanism present in (1.2). The combination of Poincare’s inequality with this entropy inequality gives rise to a linear differential inequality from which the rate of decay is established. To obtain the H” decay a family of generalized entropy inequalities for higher order spatial derivatives will be derived and combined with Poincare’s inequality. This process leads to a differential inequality in a weighted H” norm from which the exponential decay will follow. The H” decay together with a standard Sobolev inequality yields the L” decay. 2.
DECAY
RATES
FOR
SOLUTIONS
OF THE
SCALAR
EQUATIOS
In this section. we show that the solutions of the Cauchy problem for the consemation 943
law
94-t
M. E. SCHONBEK
(1.1) decay at the optimal rate of f-” I”- ’ pi m Lp and at the optimal rate of r-n’z in Lx. We will first consider the decay rate of the solutions in Lp with 1
j
i j=l In order solutions of order
to obtain the desired in a special sequence
=
p!
!
and hence
lim CJP = x. P-r
rate of decay in L”, we need to establish the decay rate for the of Lp norms with p = 2’. In this case the coefficients C, will be
that is the growth of the coefficients C, is slow enough. We now state two theorems, the first one establishes the decay rate of the solutions of (1.1) in LP for all p 2 1, the second one establishes the decay rate in LP for p = 2’ with s 2 1. The techniques used in the proof of both theorems are similar. Since the arguments for the second proof contain those needed for the first proof we shall only present the proof of the second theorem. For existence of solutions of (1.1) we refer the reader to [3]. THEOREM 2.1. Let F’ E C’, ug E L’ n C’ fl H’(R”).
u,) then for 1 6p
If u is a solution
of (1.1) with initial
data
If u is a solution
of (1.1) with initial
data
< 3~
where
THEOREM 2.2. Let F’ E Cl, u(, E L’ f7 C1 f? /ii(
u,, then for all p = 2’ with s 2 1 lL’(4LP
c A;‘” (r + 1) -n/2(1 - i/PI.
(2.1)
Here A, defined
= 2’rBhs ,:pzp (1 + ~L&~)
with
and
B
by s-l h, = 2 2i = 2’ - 1, ri = 0, rs = (s - l)(n/2
+ 1) i 2r,_, , B = 4 coo nni2,
j=O
where
h,, rr
o0 is the measure
of the n dimensional
unit sphere.
945
Uniform decay rates for parabolic conservation laws
Proof.
we let iVp = ,ma: (1 + ju Olt,). By using a linear change - -P of independent variables, one can reduce the general constant coefficient elliptic operator on the right-hand side of (1.1) to the Laplacian while preserving the general divergence structure of the equation. Therefore we shall work with the equation For notational
convenience,
u, + jt $(4 where the p will be in C’. The proof of (2.1) follows the one presented in [5].
by induction.
= Au
The first step corresponding
(2.2)
to s = 1 is similar
to
Step 1. By energy methods, we derive an equation which connects the L’ norm of the solution with the L2 norm of its gradient. An application of Plancherel’s theorem and a decomposition of the frequency domain into two time dependent subspaces yields a first order differential inequality for the L’ norm of the Fourier transform of the solution. The time dependent subspaces are an n-sphere S of radius R=
2-In 12 i t+ 11
and its complement. This decomposition allows us to bound the inhomogeneous term of the differential inequality by the volume of S yielding a new differential inequality from which the desired L’ rate of decay follows. To begin with, we multiply equation (2.2) by u and integrate by parts to obtain
(2.3) The second
integral
can be rewritten
as 2
where infinity,
qj(u)
=
Jo"
it follows
j Rn
Zd
q'(u)
dx,
axj
f’(s) ds. Recalling that u vanishes at infinity since the initial data vanishes that the second integral is equal to zero. Hence (2.3) is equal to
at
(2.4) Plancherel’s
where
theorem
yields
916
.Ll.E. SCHONBEK
Hence
Reordering
the terms
hIultiplying
both sides of the last inequality
Recalling
yields
that the solutions
by the integrating
of (1.1) satisfy an L’ contraction ILlILl c
it follows
that iris, s ‘I~,);~L. Therefore.
and integrating
the right side it follo\vs
C
:LLo I,l(f
+
1)”
Plancherel’s
theorem
j R” Recalling
yields
sphere
of radius
one.
Integrating
the last
that
jl1012dyi (t + 1))” + ~L~&co,,~“~‘(~ + l)-”
that
that
[2], i.e.
ln 1 n(r + l)_’ - -n C2r+l [
A , = Bl\’ 3 = &,&’ it follows
principle
that
again it follows
ll/’ du d (j
(t + I)” gives
IAL1
the last inequality
lvhere wO is the measure of the n-dimensional inequality kvith respect to time yields
Using
factor
max (1 + 1~~12~) lSrS2
2
(2.5)
Uniform
decay
rates for parabolic
conservation
laws
947
so that (2.5) yields
Step 2. Suppose
now that inequality
i R”
(2.1) holds for q = T1. ILL/~dx c A&
that is
+ l)-” 2(‘J-1).
(2.6)
To show that (2.1) holds with p = 2’ we need to derive an equation similar to (2.3). This equation will connect the L’ norm of the pth power of the solution with the L’ norm of the qth power of the gradient of the solution. By Plancherel’s theorem and some manipulations similar to the ones used in the case p = 2 we obtain a differential inequality for the L’ norm of the qth power of the Fourier transform of the solution. Here again the frequency domain will be divided into two time dependent subspaces: the n-sphere S, with radius I 2
and its complement S;. As in the L2 case, the inhomogeneous term will be bounded by the volume of S, in order to derive a new differential inequality from where the desired rate of decay will be obtained. We start by multiplying equation (2.2) by up-’ and integrating in space and time to obtain
dJ- lujp dx = -p zf R”
J”~4$~w~-P
j
R"
up- ’ au cl\.
R"
The same reasoning as in step one shows that the first integral on the right-hand hence after an integration by parts on the second integral. vve have
dj zf
lulpdx =
-(P
- l)P i I=1
R”
Noting
j
ll?-211;,
side vanishes,
dx.
R”
that
It follows
that
where the last inequality follows sincep(p to the last inequality yields
- l)q-’
3 1 fors > 2. Applying
Plancherel’s
theorem
hi. E. SCHONBEK
948
where
we have let w = tiq. Let S, = [g: ,s”, < ($)I’:].
We now split the integral on the right-hand side of (2.7) into an integral S; and we bound iei on S; by the radius of S, obtaining
The last inequality
is now multiplied
-$+lPjR”
/WI?
by the integrating d;‘s(t+
factor
over S, and one over
(I + 1)“q
1)"9
Hence
Integrating
the right-hand
The inductive
hypothesis
side yields
(2.6) yields the following
bound
for iwl$:
1~14dx)‘sA;(t+
Iwit < (j
I)-“(q-‘1
R” where
Ai
the bound
where
=
2?rJ-1B?h
S--I
Ni.
Noting
that 2h,_,
= h, - 1 and IV: < 4N,, it follows
by (2.8) and
for /WI: that
(Y= 2r,_ 1 + (n/2 + 1) (s - 1). Since 4n”‘* o. = B and 2/(n + 2) G 2/3 it follows
;[(i+ We now integrate
1)“q
jRn IU’112dj]
in time and use Plancherel’s
(2 + 1)"q j
theorem
1)“/2-1. to obtain
juIpdx s j juolP dx + ;A,([
R”
Since A,/3
=+A&+
- 1) it follows
(uIP d_xs A, (t + 1) -“%j R”
+ l)“?
R”
> ju o1ALPand n/2 - nq = -n/2(p
that
from (2.9) that l)
(2.9)
Uniform
This last inequality
completes
decay
rates for parabolic
the proof of theorem
conservation
laws
9-19
2.2.
COROLLARY 2.1.Let fj E C’, u. E L’ fl H’ n C’(P).
If u is a solution
of (1.1) with initial
data u. then
where A = 2”‘2C’BN,, with N, = ,mryz (1 + IuoI,_,). Proof. By theorem
2.2, we have for p = 2’ lu(t)lx
Hence
=
!i~:
(1
lulP
dx)
‘If
s
(A,)‘/P(t
!Lir
+ l)-““.
to show that lim (Ap)“p = A. For this we note that J-+X
it suffices
r, = [(s - 1) + 2(q - s)]O = [p - (s + l)]O where
q = s’-l
and 8 = n/2 + 1. Hence (AP)r/P c 2(‘-(S+ i)k’)eBr-i/P
,Tra2. (1 +
I~oM
so that lim (AP)ljP x-+x
= 2e BN,
= A.
In the next corollary we establish optimal decay rates for the solutions p 2 1. The estimate that will be obtained is of the type
of (1.1) in LP for all
s c,v,(1+ f)-n.‘311‘PI. ILl(f)lLP This result differs from the one obtained in theorem 2.1 since here the coefficient c is independent of p and the estimate depends on all the L’ norms of the initial data. not only on the ones with r sp.
COROLLARY 2.2.Let fj and u. be as in theorem
2.1. If u is a solution
of (1.1) with initial
data
uo, then
l4)l LPs 2”BN,(l + 4-“/2(‘-‘lP). Proof. The proof is a consequence of theorem 2.2, corollary 2.1 and the following Sobolev interpolation inequality for Lp spaces: Let p E (2, x) then
standard
I(P,-?),lP. I4 LPz=lui’/“l L’ uL
Applying
this inequality
in our case yields
(1w where that
p = -n/2(1
- 1/2)2/p
q’” < /#P&P-?)iP(f
+ (p - 2)/2 = -n/2(1
+
- l/p).
1)s
A simple
calculation
will show
950
E.
hl.
SCHONBEK
3. L’ RATES OF DECAY
In this section of the form
we consider
the Cauchy
problem
11, + divf((u)
FOR SYSTEMS
for systems
of parabolic
conservation
laws
= Au (3.1)
n(x, 0) = un(x) where u = U(X. t) f RP, x E R”. u. is smooth and f = (f’, . show that periodic solutions of (3.1) decay at an exponential supposed that the associated system u, + divf(u)
,fp) with?:
RP-+ R” smooth. We rate in the L’ norm. It will be
= 0
(3.2)
is hyperbolic and admits a uniformly convex entropy in the sense of Lax [4]. We recall that a pair of functions 7, q where 7: RP, q = (ql, . . , q,J and q,: RP- R is an entropy flux pair if all smooth solutions of (3.2) satisfy the additional conservation law q(u), + div q(u) = 0. The last equation
is equivalent
to the compatibility Vq .
where fl = (f; _ff, . ,f;). Condition and multiplying (3.1) by Vr],
(3.3)
condition
Tf, = vqi,
(3.4) follows by carrying
(3.3) out the differentiation
in (3.3)
n cqu,
+ c, Oq’u,, = 0
V~U,
+ C
]=I n
V17VfjU.,.
/=1
We note that if ‘I, q is an entropy flux pair for the hyperbolic of (3.1) satisfy the additional equation n qr + divq
= Aq - 2 i=
In this section satisfy
the systems
under consideration
system
(3.5)
V?qu:,. 1
will admit uniformly
V’@
(3.2) then all solutions
convex
entropies,
i.e. they
2 Si;“i?
for some S > 0. This class of systems include symmetric systems, that is systems for which the flux function f arises from a gradient of a scalar potential @, and f = V@. Here r](u) = Xuj/2 serves as an entropy [4]. The proof of the decay rate uses energy estimates in conjunction with Poincare’s inequality. Specifically, we show that if system (3.1) admits an entropy ‘I, then all smooth periodic solutions satisfy an integral inequality of the form
IK
q(u),&<
-&
(3.6) IK
Uniform
where ff > 0. The set K is defined of 11, i.e.
decay
rates for parabolic
conssrvarion
laws
by ={x: 0 c x, < L,}, L, is the period
u(x + L,e,, t) = u(x, t)
951
of the ith component
i=l.....n
(3.7)
with e, the ith element of the canonical basis of R”. e.g. ez = (0, 1, 0, .. ,0). In vvhat follows we will say that u is L-periodic if L = (L.,, . , L,) and 11satisfies (3.7). The exponential decay of solutions for systems (3.1), admitting a uniformly convex entropy rl which satisfies n(u) 3 c(u)‘, c > 0, will follow by (3.6) and Poincare’s inequality. We first establish a theorem on the decay for functions satisfying the integral inequality (3.6). We then show that solutions of (3.1) satisfy (3.6) and therefore decay. THEOREM 3.1. Let 17:RP--, R be an arbitrary uniformly convex function satisfying V(U) 5 cjc~1’, c > 0. Let x E R”. t E R,. Suppose that u = u(x, I) E RP is a smooth L-periodic function with vanishing average ti, that is
fi=h i K Ll(x, r) dx = 0. where k = {x:x = (x1.. ..,x,), 0
where
M and y depend
/KI = II;=,
L,. If u satisfies
the
integral
only on ‘7, n, L and u(x, 0).
Proof. From (3.6) and the uniform
convexity
of 17it follows that
(3.8) side of (3.7), Poincare’s inequality will be used. For completeness To bound the right-hand we recall Poincare’s inequality: Let u E R” be a L-periodic Ci function with average
then
(3.9) only on n and L. Let u = u and combine
where
p depends
where
N = p&S. By hypothesis
n 2 c[uI’, hence
(3.8) with (3.9) to obtain
if we let y = c-*N it follows
that
952
An integration
M.
E. SCHOXBEK
in time yields
I
q(u) dx C ewyr iK
K
Hence using the hypothesis
rp(“, 0) dx.
‘I 2 c/1112again yields IK
Iu/’ dr G M exp( - yt).
In the remainder of the paper we suppose that U,,(X)andfsatisfy general conditions insuring the existence of smooth solutions u of (3.1) with initial data uO(x). The next theorem and corollary show that periodic solutions of (3.1) satisfy (3.6) and hence decay exponentially. THEOREM3.2. Let PJ:RP+ R be an entropy corresponding to the system (3.1) and let u be an L-periodic smooth solution, then u satisfies inequality (3.6). Proof. Integrating
equation (3.5) in space yields
d r](u) du + divg(u)dr=j~A~d~-j~~~~~iiidr dr i K iK where K is defined as in theorem 3.1. The periodicity d V(U) du = dr i K
i
of 11yields C’~U;~ du.
i K 1=i
COROLL.&RY3.1. Let Us be a smooth L-periodic function. Let k be defined as in theorem 3.1. Suppose (3.1) admits a uniformly convex entropy q which satisfies q(u) b c/l~:~.c > 0. If u is a smooth solution of (3.1) with initial data uo(x). then [u(r) - ti/L’(Ic) 4 M exp( - yt). where 12=
& I K U(X, f) dr
and M and y depend only on llo, 17,L, and n. Proof. The uniqueness of solutions for system (3.1) and the L-periodicity of 14;)imply the L-periodicity of u. We first show that U is constant. Integrating (3.1) in space yields
%~K~~d~=j~L~,,=-~Kdivfdr+~KA~dr. The right-hand side vanishes by the periodicity of u and hence 12is constant. Without loss of generality we suppose ti = 0. If not, let u = LL - ti,then u satisfies u, + divg(u) = Au, where g(u) = f(u + ~2). Hence corollary 3.1 follows from theorems 3.1 and 3.2.
Uniform decay rates for parabolic conservation laws
953
Remark. The decay results obtained for periodic solutions of (3.1) are also valid for solutions in a bounded domain with zero boundary data. Example 1. In [4], Lax constructs strictly convex entropies for 2 x 2 systems. Hence corollary 3.1 can be applied to 2 x 2 systems which have solutions with a known L” a priori bound. Example 2. Navier-Stokes
equations. div I( = 0 u, + (u . V)u = -VP + vhu
v>o
(3.10)
u(x, 0) = &l(x) where u = u(x, t) E R” , x E R”, t E R,. The functions Us and p(x, t) are smooth and Lperiodic. In the case n = 2, it is known that system (3.10) has smooth solutions. It is well known that such solutions satisfy the integral inequality (3.6) with
specifically they satisfy [ Iu/2 dx = -2v An easy computation
/ jVul’ dw
(3.11)
shows that
li=&
J
K u(x, t) dx = constant.
Hence from theorem 3.1 and (3.11) the exponential
decay follows, that is
lu(x, t) - ~21~2C M exp( - yt) where M, y depend only on uo. We note finally that under sufficient assumptions of smoothness for solutions of (3.10) in higher dimension, theorem (3.1) will give their exponential decay. 4. H” AND L’ RATES OF DECAY
In this section we show that periodic solutions of the system (3.1) decay exponentially in H” and in L”. The H” decay is obtained by combining certain generalized entropy inequalities for the higher order spatial derivatives of the solutions and Poincare’s inequality. This leads to a linear differential inequality connecting a weighted H”’ norm of the solution with an m + 1 weighted norm from where the exponential decay follows. The L” decay is obtained from the H”’ decay and a standard Sobolev inequality. Letf= (f’,. . .,f’),fj: R” + R. The following notation will be used.
where lfilz and If’l%are the usual L2 and L” norms. Let r E N and CY= (cY,, . . . , an), q E N,
954
&I.
E.
SCHONBEK
then we let
where INi = i: cr, 1=1
D”=D,;=fiD;;.
and
I=1
THEOREM 4.1. LetfE C+‘(IC’, R”). Let ll0 E C(RP) be a L-periodic function. Suppose (3.2) admits a uniformly convex entropy ‘I. satisfying q(u) 2 cu’, c > 0. If 11is a solution of (3.1) with initial data u. then
and if m > [n/2] then
x
r
where
M and y depend
Proof.
We introduce
lJD’u(l,
c Mexp(-yt),
I]
only on n, m. f and uo. the following
weighted
H’ norms
(u(t)); = i. k,,, JID'ull;s 2 0.
(-1.1)
quantities where k. o = 0, /c,,~ = 0 and the weights k,,, with s 2 1, 1 s r < s are nonnegative defined below by induction. The main step in the proof is to establish a differential inequality of the type (3.6) for the norms (4.1). Specifically we obtain an inequality connecting the rate of change in time of the s-norm with the negative of an s -I- 1 norm, that is we show (4.2) where c indicates here and below a positive constant depending only on ‘I, n, f.IL,) and the L” norm of ~1. Once (4.2) is established the H” decay will follow by Poincare’s inequality. To establish (4.2) we proceed by induction on s. For s = 0 the equality is reduced to
and was established in theorem 3.1 with k,, = CYSand c = 1. We now suppose that (4.2) holds for all integers less than s and show that it holds for s. Taking the D” derivative of (3.1) with INI = s and multiplying by D”u it follows that DLudDnuj
+ D”dD”
div f (u) = D”dD”Aul
Uniform
forj=
1,. . . . p. An integration
decay
rates for parabolic
955
laws
in space yields
Integrating by parts in the second hand side gives
integral
& = - i I=1
where Schwarz’s inequality with 1cy\ = s yields
conservation
on the left-hand
j
side and in the integral
of the right-
jDn+e,l(‘j2 K
was used in the last step. Hence
adding
all derivatives
of order
N,
of the Sobolev
embedding
Hence
We will need the following theorem, [2].
lemma which is a well known consequence
LEMMA 4.1. Let u = (u’, f(0) = 0. Then ifs 2 1
. , up), uJ E Cs(R,),
; = 1,
. . , p. Suppose
f= f(rt) f C’(RP. R”),
(4.4)
IlD”f(~)ll~ s c]]Ds~& 3 where
c depends
only on s.
f
and the L” norm
Remark 1. We note that lemma in R” and dR is in C’[l].
of
K.
4.1 is also valid if the U! E C(Q)
Remark 2. Without loss of generality we can suppose f(0) = 0 otherwise we replace f by g = f - f(0). Combining inequalities (4.3) and (4.4) yields
with R a bounded
that the flux function
$‘sull; 6 - +llDscl~+ + cIID~KI~;.
domain
f in (4.1) satisfies
(4.5)
M. E. SCHONBEK
956
By inductive
hypothesis
we have that
(4.6) Now we define the k,,,. 0 < r < s and k,,,,, inequality (4.2). That is let
so that
the result
k,., = k,,,-i
O
k,,,
k J-f I.3 = &k,,,.
= $k,,,_,
of adding
(4.5)
and (1.6)
is
Hence adding (4.5) and (4.6) gives inequality (4.2). Since u is periodic and satisfies (3.1). the average
vanishes
Combining
for all N. Hence
this inequality
By hypothesis
Poincare’s
inequality
yields
with (1.2) will give
we had V(U) 2 CU?, thus
where &,I = Ik,,,, k,
= k+ 1.5
O
The constant ? can be chosen positive since the k,,,, 1 S r S s are positive. Integrating inequality (4.7) yields the exponential decay of the (I), and hence the exponential decay of the W norms follows. The L” decay is obtained from the decay in W’ and the Sobolev inequality for r < m - [n/2].
REFERENCES 1. FRIEDMAS A.. Parrial Differenrial Equations, Robert E. Krieger, Huntington, New York (1976). 2. KLMSERMW S.. Global existence for nonlinear wave equations, CPA.M XxX111, 43-101 (1980). 3. KOTLAW D. B.. Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data. J. math. Analysis Applic. 35, 563-576 (1971). 4. LAX P., Shock waves and entropy, in Contributions to Nonlinear Function Analysis, (Edited by Z.ARA_\TOSELLO), Academic Press. New York (1971). 5. SCHOXBEK M., Decay of solutions to parabolic conservation laws, Communs parrial dif” Eqns 7, 449173 (1980).
DECAY
RATES
FOR PARABOLIC MARIA
Department
of Mathematics.
Duke
(Received in reoised form 17 ,Vouember Key words and phrases: Parabolic
conservation
E.
CONSERVATION
LAWS
SCHONBEK
University.
Durham,
NC 27706.
U.S.A.
1983; received for publication 21 October 1985) laws, decay
rates.
1. INTRODUCTION
paper we establish optimal uniform rates of decay for solutions of the Cauchy problem with large data to scalar parabolic conservation laws in several space dimensions, IN THIS
where the II,, are constants, and to parabolic systems of conservation
laws
$ + divf(L1) = Au,
(1.2)
where u = U(X, t) E RP, x E R” and f, = (f’, . . . ,fp) with f: RF- R”. Equations of this type arise in continuum mechanics where they serve as models for elastic solids and ideal gases. In an earlier paper [j], we have shown that solutions of the scalar equation (1.1) decay in L’ and L’ with a rate of ten”. It is not difficult to verify that t -n:’ is the optimal rate in L’. On the other hand, one can expect the faster rate of ten,’ m ’ L” based on an analysis of the heat equation. In Section 2, we shall show that solutions of the scalar equation (1.1) decay at the optimal rates t -n “‘-‘;P) in LP for 1 < p < x and at the optimal rate of t-“/I in L”. In order to establish the decay rate we derive an entropy inequality for the function up. Applying Fourier analysis yields an inequality for the Fourier transform of an appropriate power depending on p. A subdivision of the frequency domain into two time dependent subspaces leads to the desired decay rate. In Sections 3 and 4 we show that periodic solutions of system (1.2) decay exponentially in H” and L” for a class of general flux functionsfi With regard to background we recall that decay without a rate of solutions of (1.2) in one spatial dimension was established in [5]. In the present paper we first obtain the exponential decay in the L?-norm. We derive an entropy inequality which expresses the dissipative mechanism present in (1.2). The combination of Poincare’s inequality with this entropy inequality gives rise to a linear differential inequality from which the rate of decay is established. To obtain the H” decay a family of generalized entropy inequalities for higher order spatial derivatives will be derived and combined with Poincare’s inequality. This process leads to a differential inequality in a weighted H” norm from which the exponential decay will follow. The H” decay together with a standard Sobolev inequality yields the L” decay. 2.
DECAY
RATES
FOR
SOLUTIONS
OF THE
SCALAR
EQUATIOS
In this section. we show that the solutions of the Cauchy problem for the consemation 943
law
94-t
M. E. SCHONBEK
(1.1) decay at the optimal rate of f-” I”- ’ pi m Lp and at the optimal rate of r-n’z in Lx. We will first consider the decay rate of the solutions in Lp with 1
j
i j=l In order solutions of order
to obtain the desired in a special sequence
=
p!
!
and hence
lim CJP = x. P-r
rate of decay in L”, we need to establish the decay rate for the of Lp norms with p = 2’. In this case the coefficients C, will be
that is the growth of the coefficients C, is slow enough. We now state two theorems, the first one establishes the decay rate of the solutions of (1.1) in LP for all p 2 1, the second one establishes the decay rate in LP for p = 2’ with s 2 1. The techniques used in the proof of both theorems are similar. Since the arguments for the second proof contain those needed for the first proof we shall only present the proof of the second theorem. For existence of solutions of (1.1) we refer the reader to [3]. THEOREM 2.1. Let F’ E C’, ug E L’ n C’ fl H’(R”).
u,) then for 1 6p
If u is a solution
of (1.1) with initial
data
If u is a solution
of (1.1) with initial
data
< 3~
where
THEOREM 2.2. Let F’ E Cl, u(, E L’ f7 C1 f? /ii(
u,, then for all p = 2’ with s 2 1 lL’(4LP
c A;‘” (r + 1) -n/2(1 - i/PI.
(2.1)
Here A, defined
= 2’rBhs ,:pzp (1 + ~L&~)
with
and
B
by s-l h, = 2 2i = 2’ - 1, ri = 0, rs = (s - l)(n/2
+ 1) i 2r,_, , B = 4 coo nni2,
j=O
where
h,, rr
o0 is the measure
of the n dimensional
unit sphere.
945
Uniform decay rates for parabolic conservation laws
Proof.
we let iVp = ,ma: (1 + ju Olt,). By using a linear change - -P of independent variables, one can reduce the general constant coefficient elliptic operator on the right-hand side of (1.1) to the Laplacian while preserving the general divergence structure of the equation. Therefore we shall work with the equation For notational
convenience,
u, + jt $(4 where the p will be in C’. The proof of (2.1) follows the one presented in [5].
by induction.
= Au
The first step corresponding
(2.2)
to s = 1 is similar
to
Step 1. By energy methods, we derive an equation which connects the L’ norm of the solution with the L2 norm of its gradient. An application of Plancherel’s theorem and a decomposition of the frequency domain into two time dependent subspaces yields a first order differential inequality for the L’ norm of the Fourier transform of the solution. The time dependent subspaces are an n-sphere S of radius R=
2-In 12 i t+ 11
and its complement. This decomposition allows us to bound the inhomogeneous term of the differential inequality by the volume of S yielding a new differential inequality from which the desired L’ rate of decay follows. To begin with, we multiply equation (2.2) by u and integrate by parts to obtain
(2.3) The second
integral
can be rewritten
as 2
where infinity,
qj(u)
=
Jo"
it follows
j Rn
Zd
q'(u)
dx,
axj
f’(s) ds. Recalling that u vanishes at infinity since the initial data vanishes that the second integral is equal to zero. Hence (2.3) is equal to
at
(2.4) Plancherel’s
where
theorem
yields
916
.Ll.E. SCHONBEK
Hence
Reordering
the terms
hIultiplying
both sides of the last inequality
Recalling
yields
that the solutions
by the integrating
of (1.1) satisfy an L’ contraction ILlILl c
it follows
that iris, s ‘I~,);~L. Therefore.
and integrating
the right side it follo\vs
C
:LLo I,l(f
+
1)”
Plancherel’s
theorem
j R” Recalling
yields
sphere
of radius
one.
Integrating
the last
that
jl1012dyi (t + 1))” + ~L~&co,,~“~‘(~ + l)-”
that
that
[2], i.e.
ln 1 n(r + l)_’ - -n C2r+l [
A , = Bl\’ 3 = &,&’ it follows
principle
that
again it follows
ll/’ du d (j
(t + I)” gives
IAL1
the last inequality
lvhere wO is the measure of the n-dimensional inequality kvith respect to time yields
Using
factor
max (1 + 1~~12~) lSrS2
2
(2.5)
Uniform
decay
rates for parabolic
conservation
laws
947
so that (2.5) yields
Step 2. Suppose
now that inequality
i R”
(2.1) holds for q = T1. ILL/~dx c A&
that is
+ l)-” 2(‘J-1).
(2.6)
To show that (2.1) holds with p = 2’ we need to derive an equation similar to (2.3). This equation will connect the L’ norm of the pth power of the solution with the L’ norm of the qth power of the gradient of the solution. By Plancherel’s theorem and some manipulations similar to the ones used in the case p = 2 we obtain a differential inequality for the L’ norm of the qth power of the Fourier transform of the solution. Here again the frequency domain will be divided into two time dependent subspaces: the n-sphere S, with radius I 2
and its complement S;. As in the L2 case, the inhomogeneous term will be bounded by the volume of S, in order to derive a new differential inequality from where the desired rate of decay will be obtained. We start by multiplying equation (2.2) by up-’ and integrating in space and time to obtain
dJ- lujp dx = -p zf R”
J”~4$~w~-P
j
R"
up- ’ au cl\.
R"
The same reasoning as in step one shows that the first integral on the right-hand hence after an integration by parts on the second integral. vve have
dj zf
lulpdx =
-(P
- l)P i I=1
R”
Noting
j
ll?-211;,
side vanishes,
dx.
R”
that
It follows
that
where the last inequality follows sincep(p to the last inequality yields
- l)q-’
3 1 fors > 2. Applying
Plancherel’s
theorem
hi. E. SCHONBEK
948
where
we have let w = tiq. Let S, = [g: ,s”, < ($)I’:].
We now split the integral on the right-hand side of (2.7) into an integral S; and we bound iei on S; by the radius of S, obtaining
The last inequality
is now multiplied
-$+lPjR”
/WI?
by the integrating d;‘s(t+
factor
over S, and one over
(I + 1)“q
1)"9
Hence
Integrating
the right-hand
The inductive
hypothesis
side yields
(2.6) yields the following
bound
for iwl$:
1~14dx)‘sA;(t+
Iwit < (j
I)-“(q-‘1
R” where
Ai
the bound
where
=
2?rJ-1B?h
S--I
Ni.
Noting
that 2h,_,
= h, - 1 and IV: < 4N,, it follows
by (2.8) and
for /WI: that
(Y= 2r,_ 1 + (n/2 + 1) (s - 1). Since 4n”‘* o. = B and 2/(n + 2) G 2/3 it follows
;[(i+ We now integrate
1)“q
jRn IU’112dj]
in time and use Plancherel’s
(2 + 1)"q j
theorem
1)“/2-1. to obtain
juIpdx s j juolP dx + ;A,([
R”
Since A,/3
=+A&+
- 1) it follows
(uIP d_xs A, (t + 1) -“%j R”
+ l)“?
R”
> ju o1ALPand n/2 - nq = -n/2(p
that
from (2.9) that l)
(2.9)
Uniform
This last inequality
completes
decay
rates for parabolic
the proof of theorem
conservation
laws
9-19
2.2.
COROLLARY 2.1.Let fj E C’, u. E L’ fl H’ n C’(P).
If u is a solution
of (1.1) with initial
data u. then
where A = 2”‘2C’BN,, with N, = ,mryz (1 + IuoI,_,). Proof. By theorem
2.2, we have for p = 2’ lu(t)lx
Hence
=
!i~:
(1
lulP
dx)
‘If
s
(A,)‘/P(t
!Lir
+ l)-““.
to show that lim (Ap)“p = A. For this we note that J-+X
it suffices
r, = [(s - 1) + 2(q - s)]O = [p - (s + l)]O where
q = s’-l
and 8 = n/2 + 1. Hence (AP)r/P c 2(‘-(S+ i)k’)eBr-i/P
,Tra2. (1 +
I~oM
so that lim (AP)ljP x-+x
= 2e BN,
= A.
In the next corollary we establish optimal decay rates for the solutions p 2 1. The estimate that will be obtained is of the type
of (1.1) in LP for all
s c,v,(1+ f)-n.‘311‘PI. ILl(f)lLP This result differs from the one obtained in theorem 2.1 since here the coefficient c is independent of p and the estimate depends on all the L’ norms of the initial data. not only on the ones with r sp.
COROLLARY 2.2.Let fj and u. be as in theorem
2.1. If u is a solution
of (1.1) with initial
data
uo, then
l4)l LPs 2”BN,(l + 4-“/2(‘-‘lP). Proof. The proof is a consequence of theorem 2.2, corollary 2.1 and the following Sobolev interpolation inequality for Lp spaces: Let p E (2, x) then
standard
I(P,-?),lP. I4 LPz=lui’/“l L’ uL
Applying
this inequality
in our case yields
(1w where that
p = -n/2(1
- 1/2)2/p
q’” < /#P&P-?)iP(f
+ (p - 2)/2 = -n/2(1
+
- l/p).
1)s
A simple
calculation
will show
950
E.
hl.
SCHONBEK
3. L’ RATES OF DECAY
In this section of the form
we consider
the Cauchy
problem
11, + divf((u)
FOR SYSTEMS
for systems
of parabolic
conservation
laws
= Au (3.1)
n(x, 0) = un(x) where u = U(X. t) f RP, x E R”. u. is smooth and f = (f’, . show that periodic solutions of (3.1) decay at an exponential supposed that the associated system u, + divf(u)
,fp) with?:
RP-+ R” smooth. We rate in the L’ norm. It will be
= 0
(3.2)
is hyperbolic and admits a uniformly convex entropy in the sense of Lax [4]. We recall that a pair of functions 7, q where 7: RP, q = (ql, . . , q,J and q,: RP- R is an entropy flux pair if all smooth solutions of (3.2) satisfy the additional conservation law q(u), + div q(u) = 0. The last equation
is equivalent
to the compatibility Vq .
where fl = (f; _ff, . ,f;). Condition and multiplying (3.1) by Vr],
(3.3)
condition
Tf, = vqi,
(3.4) follows by carrying
(3.3) out the differentiation
in (3.3)
n cqu,
+ c, Oq’u,, = 0
V~U,
+ C
]=I n
V17VfjU.,.
/=1
We note that if ‘I, q is an entropy flux pair for the hyperbolic of (3.1) satisfy the additional equation n qr + divq
= Aq - 2 i=
In this section satisfy
the systems
under consideration
system
(3.5)
V?qu:,. 1
will admit uniformly
V’@
(3.2) then all solutions
convex
entropies,
i.e. they
2 Si;“i?
for some S > 0. This class of systems include symmetric systems, that is systems for which the flux function f arises from a gradient of a scalar potential @, and f = V@. Here r](u) = Xuj/2 serves as an entropy [4]. The proof of the decay rate uses energy estimates in conjunction with Poincare’s inequality. Specifically, we show that if system (3.1) admits an entropy ‘I, then all smooth periodic solutions satisfy an integral inequality of the form
IK
q(u),&<
-&
(3.6) IK
Uniform
where ff > 0. The set K is defined of 11, i.e.
decay
rates for parabolic
conssrvarion
laws
by ={x: 0 c x, < L,}, L, is the period
u(x + L,e,, t) = u(x, t)
951
of the ith component
i=l.....n
(3.7)
with e, the ith element of the canonical basis of R”. e.g. ez = (0, 1, 0, .. ,0). In vvhat follows we will say that u is L-periodic if L = (L.,, . , L,) and 11satisfies (3.7). The exponential decay of solutions for systems (3.1), admitting a uniformly convex entropy rl which satisfies n(u) 3 c(u)‘, c > 0, will follow by (3.6) and Poincare’s inequality. We first establish a theorem on the decay for functions satisfying the integral inequality (3.6). We then show that solutions of (3.1) satisfy (3.6) and therefore decay. THEOREM 3.1. Let 17:RP--, R be an arbitrary uniformly convex function satisfying V(U) 5 cjc~1’, c > 0. Let x E R”. t E R,. Suppose that u = u(x, I) E RP is a smooth L-periodic function with vanishing average ti, that is
fi=h i K Ll(x, r) dx = 0. where k = {x:x = (x1.. ..,x,), 0
where
M and y depend
/KI = II;=,
L,. If u satisfies
the
integral
only on ‘7, n, L and u(x, 0).
Proof. From (3.6) and the uniform
convexity
of 17it follows that
(3.8) side of (3.7), Poincare’s inequality will be used. For completeness To bound the right-hand we recall Poincare’s inequality: Let u E R” be a L-periodic Ci function with average
then
(3.9) only on n and L. Let u = u and combine
where
p depends
where
N = p&S. By hypothesis
n 2 c[uI’, hence
(3.8) with (3.9) to obtain
if we let y = c-*N it follows
that
952
An integration
M.
E. SCHOXBEK
in time yields
I
q(u) dx C ewyr iK
K
Hence using the hypothesis
rp(“, 0) dx.
‘I 2 c/1112again yields IK
Iu/’ dr G M exp( - yt).
In the remainder of the paper we suppose that U,,(X)andfsatisfy general conditions insuring the existence of smooth solutions u of (3.1) with initial data uO(x). The next theorem and corollary show that periodic solutions of (3.1) satisfy (3.6) and hence decay exponentially. THEOREM3.2. Let PJ:RP+ R be an entropy corresponding to the system (3.1) and let u be an L-periodic smooth solution, then u satisfies inequality (3.6). Proof. Integrating
equation (3.5) in space yields
d r](u) du + divg(u)dr=j~A~d~-j~~~~~iiidr dr i K iK where K is defined as in theorem 3.1. The periodicity d V(U) du = dr i K
i
of 11yields C’~U;~ du.
i K 1=i
COROLL.&RY3.1. Let Us be a smooth L-periodic function. Let k be defined as in theorem 3.1. Suppose (3.1) admits a uniformly convex entropy q which satisfies q(u) b c/l~:~.c > 0. If u is a smooth solution of (3.1) with initial data uo(x). then [u(r) - ti/L’(Ic) 4 M exp( - yt). where 12=
& I K U(X, f) dr
and M and y depend only on llo, 17,L, and n. Proof. The uniqueness of solutions for system (3.1) and the L-periodicity of 14;)imply the L-periodicity of u. We first show that U is constant. Integrating (3.1) in space yields
%~K~~d~=j~L~,,=-~Kdivfdr+~KA~dr. The right-hand side vanishes by the periodicity of u and hence 12is constant. Without loss of generality we suppose ti = 0. If not, let u = LL - ti,then u satisfies u, + divg(u) = Au, where g(u) = f(u + ~2). Hence corollary 3.1 follows from theorems 3.1 and 3.2.
Uniform decay rates for parabolic conservation laws
953
Remark. The decay results obtained for periodic solutions of (3.1) are also valid for solutions in a bounded domain with zero boundary data. Example 1. In [4], Lax constructs strictly convex entropies for 2 x 2 systems. Hence corollary 3.1 can be applied to 2 x 2 systems which have solutions with a known L” a priori bound. Example 2. Navier-Stokes
equations. div I( = 0 u, + (u . V)u = -VP + vhu
v>o
(3.10)
u(x, 0) = &l(x) where u = u(x, t) E R” , x E R”, t E R,. The functions Us and p(x, t) are smooth and Lperiodic. In the case n = 2, it is known that system (3.10) has smooth solutions. It is well known that such solutions satisfy the integral inequality (3.6) with
specifically they satisfy [ Iu/2 dx = -2v An easy computation
/ jVul’ dw
(3.11)
shows that
li=&
J
K u(x, t) dx = constant.
Hence from theorem 3.1 and (3.11) the exponential
decay follows, that is
lu(x, t) - ~21~2C M exp( - yt) where M, y depend only on uo. We note finally that under sufficient assumptions of smoothness for solutions of (3.10) in higher dimension, theorem (3.1) will give their exponential decay. 4. H” AND L’ RATES OF DECAY
In this section we show that periodic solutions of the system (3.1) decay exponentially in H” and in L”. The H” decay is obtained by combining certain generalized entropy inequalities for the higher order spatial derivatives of the solutions and Poincare’s inequality. This leads to a linear differential inequality connecting a weighted H”’ norm of the solution with an m + 1 weighted norm from where the exponential decay follows. The L” decay is obtained from the H”’ decay and a standard Sobolev inequality. Letf= (f’,. . .,f’),fj: R” + R. The following notation will be used.
where lfilz and If’l%are the usual L2 and L” norms. Let r E N and CY= (cY,, . . . , an), q E N,
954
&I.
E.
SCHONBEK
then we let
where INi = i: cr, 1=1
D”=D,;=fiD;;.
and
I=1
THEOREM 4.1. LetfE C+‘(IC’, R”). Let ll0 E C(RP) be a L-periodic function. Suppose (3.2) admits a uniformly convex entropy ‘I. satisfying q(u) 2 cu’, c > 0. If 11is a solution of (3.1) with initial data u. then
and if m > [n/2] then
x
r
where
M and y depend
Proof.
We introduce
lJD’u(l,
c Mexp(-yt),
I]
only on n, m. f and uo. the following
weighted
H’ norms
(u(t)); = i. k,,, JID'ull;s 2 0.
(-1.1)
quantities where k. o = 0, /c,,~ = 0 and the weights k,,, with s 2 1, 1 s r < s are nonnegative defined below by induction. The main step in the proof is to establish a differential inequality of the type (3.6) for the norms (4.1). Specifically we obtain an inequality connecting the rate of change in time of the s-norm with the negative of an s -I- 1 norm, that is we show (4.2) where c indicates here and below a positive constant depending only on ‘I, n, f.IL,) and the L” norm of ~1. Once (4.2) is established the H” decay will follow by Poincare’s inequality. To establish (4.2) we proceed by induction on s. For s = 0 the equality is reduced to
and was established in theorem 3.1 with k,, = CYSand c = 1. We now suppose that (4.2) holds for all integers less than s and show that it holds for s. Taking the D” derivative of (3.1) with INI = s and multiplying by D”u it follows that DLudDnuj
+ D”dD”
div f (u) = D”dD”Aul
Uniform
forj=
1,. . . . p. An integration
decay
rates for parabolic
955
laws
in space yields
Integrating by parts in the second hand side gives
integral
& = - i I=1
where Schwarz’s inequality with 1cy\ = s yields
conservation
on the left-hand
j
side and in the integral
of the right-
jDn+e,l(‘j2 K
was used in the last step. Hence
adding
all derivatives
of order
N,
of the Sobolev
embedding
Hence
We will need the following theorem, [2].
lemma which is a well known consequence
LEMMA 4.1. Let u = (u’, f(0) = 0. Then ifs 2 1
. , up), uJ E Cs(R,),
; = 1,
. . , p. Suppose
f= f(rt) f C’(RP. R”),
(4.4)
IlD”f(~)ll~ s c]]Ds~& 3 where
c depends
only on s.
f
and the L” norm
Remark 1. We note that lemma in R” and dR is in C’[l].
of
K.
4.1 is also valid if the U! E C(Q)
Remark 2. Without loss of generality we can suppose f(0) = 0 otherwise we replace f by g = f - f(0). Combining inequalities (4.3) and (4.4) yields
with R a bounded
that the flux function
$‘sull; 6 - +llDscl~+ + cIID~KI~;.
domain
f in (4.1) satisfies
(4.5)
M. E. SCHONBEK
956
By inductive
hypothesis
we have that
(4.6) Now we define the k,,,. 0 < r < s and k,,,,, inequality (4.2). That is let
so that
the result
k,., = k,,,-i
O
k,,,
k J-f I.3 = &k,,,.
= $k,,,_,
of adding
(4.5)
and (1.6)
is
Hence adding (4.5) and (4.6) gives inequality (4.2). Since u is periodic and satisfies (3.1). the average
vanishes
Combining
for all N. Hence
this inequality
By hypothesis
Poincare’s
inequality
yields
with (1.2) will give
we had V(U) 2 CU?, thus
where &,I = Ik,,,, k,
= k+ 1.5
O
The constant ? can be chosen positive since the k,,,, 1 S r S s are positive. Integrating inequality (4.7) yields the exponential decay of the (I), and hence the exponential decay of the W norms follows. The L” decay is obtained from the decay in W’ and the Sobolev inequality for r < m - [n/2].
REFERENCES 1. FRIEDMAS A.. Parrial Differenrial Equations, Robert E. Krieger, Huntington, New York (1976). 2. KLMSERMW S.. Global existence for nonlinear wave equations, CPA.M XxX111, 43-101 (1980). 3. KOTLAW D. B.. Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data. J. math. Analysis Applic. 35, 563-576 (1971). 4. LAX P., Shock waves and entropy, in Contributions to Nonlinear Function Analysis, (Edited by Z.ARA_\TOSELLO), Academic Press. New York (1971). 5. SCHOXBEK M., Decay of solutions to parabolic conservation laws, Communs parrial dif” Eqns 7, 449173 (1980).