Compactness of Solutions for Scalar Viscous Conservation Laws in Noncylindrical Domains

Transcription

1 Submetido para TEMA Compactness of Solutions for Scalar Viscous Conservation Laws in Noncylindrical Domains WLADIM NEVES 1, Instituto de Matemática, Universidade do Brasil - UFRJ C. Postal 68530, Rio de Janeiro, RJ , Brazil We utilize the kinetic formulation approach to study the compactness property for the family {u ε } ε>0, solutions of the initial-boundary value problem for the scalar viscous conservation law u ε t + div x f(u ε ) = ε x u ε in a noncylindrical domain. Considering f in C 3 and satisfying the non-degeneracy condition, we prove that u ε is compact in L 1. 1 wladimir@im.ufrj.br

2 1. Introduction Usually the partial differential equations which we observe have been guided by the perspective of Continuum Mechanics, almost obtained by systems of integral equations combining balance laws with constitutive relations. An alternative approach is the kinetic formulation, motivated by the classical kinetic theory of gases. In this approach, the state of a gas is described by the density function f(t, x, v) where t is the time, x the position and v the molecular veity. The evolution of f(t, x, v) is governed by the Boltzmann equation, which monitors the changes in the distributional molecular veities due to collisions and transport. The connections between the kinetic theory and the continuum approach are established by identifying intensive quantities like veity, density, Cauchy stress tensor, temperature, etc., with appropriate moments of the density function f(t, x, v) and showing that these fields satisfy the balance laws of continuum physics. In this context of kinetic theory, however establish in an artificial way in which f(t, x, v) is allowed to take also negative values, we prove that the family {u ε } ε>0 solutions to the problem (1.1)-(1.3), i.e. an initial-boundary value problem for scalar viscous conservation laws, is compact in L 1 (Q T ). A usual procedure to obtain this result is to derive uniform estimates (with respect to the parameter ε > 0) on t u ε L 1 (Q T ) and x u ε L 1 (Q T ). However, it seems impossible to derive such estimates for noncylindrical domains, even in the one-dimensional case, see Neves [9]. We cite principally the Lions, Perthame & Tadmor [8] paper, who introduced the kinetic formulation and study the Cauchy problem for scalar conservation laws. Here, for the main result we follow Chen & Frid [1]. The reader could also see a complete reference to the kinetic theory in Perthame [10]. Let f C 3 (; n ) be a given map, Q T an open noncylindrical smooth domain of n+1 whose points are denoted by (t, x) n. We will denote Γ T the lateral boundary of Q T and by Ω the set Q T {t = 0} =. Therefore for some T > 0, we have Q T = 0<t<T Ω t {t}, Γ T = 2 0<t<T Ω t {t}.

3 Scalar Viscous Conservation Laws 3 For each ε > 0, we consider the initial-boundary value problem u ε t + div x f(u ε ) = ε x u ε in Q T, (1.1) u ε = u 0 in Ω, (1.2) u ε = u b on Γ T. (1.3) We assume that the initial-boundary data satisfies u 0 H 1 (Ω) L (Ω; L n ), u b L 2 (0, T ; H 3/2 ( Ω t )) H 3/4 (0, T ; L 2 ( Ω t )) L (Γ T ; H n ), where H s denotes the s-dimensional Hausdorff measure and L n the n-dimensional Lebesgue measure. The initial-boundary value problem (1.1)-(1.3) has a unique solution u ε L 2 (0, T ; H 2 (Ω t )) C(0, T ; H 1 (Ω t )) L (Q T ), u ε t L 2 (Q T ). This solution satisfies sup ε>0 u ε (t, x) < and sup ε>0 D ε x u ε (t, x) 2 dx dt <, where D Q T is any compact set, see Neves [9]. The solution u ε of (1.1) may exhibit boundary layers near Γ T for ε > 0 small and appropriate boundary conditions. For instance, this happens when we try to solve the initial-boundary value problem to the scalar conservation law u t + div x f(u) = 0, letting ε 0 +, see Serre [12] Vol Equivalence with the Kinetic Formulation We begin defining the density function f. Let χ : {1, 1, 0}, +1, if 0 v < λ χ(λ, v) χ λ (v) := 1, if λ v < 0 0, otherwise. Moreover, we have χ λ (v) = [ sgn(λ v)+ sgn(v)]/2. Let u(t, x) be a function defined in Q T, and for v, we define We observe that, f(t, x, v) := χ u(t,x) (v). (2.1) u(t, x) = f(t, x, v) dv. (2.2) Now, we consider the following linear partial differential equation t f + a(v) x f x f = v m in Q T (2.3)

4 4 Wladimir Neves for some nonnegative Radon measure m(t, x, v), where a(v) = f (v). The equation (2.3) is derived from the kinetic formulation as introduced in Lions, Perthame & Tadmor [8]. In this section, for simplicity we drop the parameter ε > 0. The next lemma shows that (1.1) and (2.3) are equivalent. Lemma 2.1. Let u(t, x) be a function defined in Q T n+1, and f(t, x, v) be defined by (2.1). Then, u satisfies (1.1) in D (Q T ) if and only if f satisfies (2.3) in D (Q T ). Proof. 1. First, with u(t, x) a solution for (1.1), we show that f satisfies (2.3). For each v, let F (λ, v) = (η(λ, v), q(λ, v)) be a C 2 convex entropy pair. Then multiplying (1.1) by η (u(t, x), v), due that η is convex and the compatibility condition, i.e. q = η f, we obtain For convenience, we denote t η(u, v) + div x q(u, v) x η(u, v) 0 in D (Q T ). (2.4) L = t η(u, v) + div x q(u, v) x η(u, v) and, for any non-negative function ψ C 0 (Q T ), we have L(ψ) 0. (2.5) Now as a consequence of the Schwartz lemma on negative distributions, see Schwartz [11], there exist a non-negative Radon measure m over Q T (moreover, uniformly bounded in case of m ε ), such that for any function φ C0 (Q T ) L(φ) = φ dm. Q T As usual, we do not distinguish between L and m, i.e., L(φ) < L, φ >=< m, φ >= φ dm. Q T Despite having taken η C 2, by approximation the above result is valid for the following entropy pair Then, we obtain η(λ, v) = 1 2 [ λ v v ], q i (λ, v) = λ 0 a i(ξ) ξ η(ξ, v) dξ (i = 1,..., n). t [ u ε v v ] + div x [(f(u ε ) f(v)) sgn(u ε v) +(f(0) f(v)) sgn(v)] ε x [ u ε v v ] = 2m ε. (2.6)

5 Scalar Viscous Conservation Laws 5 Now, we differentiate (2.6) with respect to v in the sense of distributions t [ sgn(u v) + sgn(v)] div x a(v)[ sgn(u v) + sgn(v)] + x [ sgn(u v) + sgn(v)] = 2 v m in D (Q T ). Hence the result follows, that is, f satisfyies (2.3). 2. Now, if f satisfies (2.3) then we prove that η(u) satisfies (2.4). Let η be a convex smooth entropy, at least a generalized one. Multiplying (2.3) by η (v), we get δ v=u η (v) t u + δ v=u η (v)f (v) x u div x [δ v=u η (v) x u] = η (v) v m, where δ λ is the Dirac measure concentrated at λ. Moreover, taking η sublinear and integrating the above equation in with respect to v, we obtain t η(u) + div x q(u) x η(u) = η (v) m dv in D (Q T ). Since η is convex and m is a non-negative Radon measure, the result follows. 3. Compactness of {u ε } In this section we are going to prove the L 1 compactness of the family {uε } ε>0. Theorem 3.1. Let f C 3 (; n ) and satisfying the non-degeneracy condition, i.e., L 1 ({v ; τ + a(v) k = 0}) = 0, for all (τ, k) n with τ 2 + k 2 = 1. Then the family {u ε } ε>0, solutions of (1.1)-(1.3), is compact in L 1 (Q T ). Proof. 1. For each u ε (t, x) and v, we set f ε (t, x, v) = χ u ε (t,x)(v). By (2.2) it is enough to prove that f ε (t, x, v) dv is compact in L 1 (Q T ). (3.1) Moreover, we observe that f ε (t, x, v) 0 when v R 0, for any R 0 greater than the uniform boundness of u ε in L (Q T ). Hence the integral in (3.1) can be taken in a finite and fix interval ( R 0, R 0 ). 2. For each v ( R 0, R 0 ), let F (λ, v) = (η(λ, v), q(λ, v)) be an entropy pair, C 2 at least. From the proof of Lemma 2.1, we have div t,x F (u ε (t, x), v) = ε x η(u ε (t, x), v) m ε. (3.2) Since u ε is uniformly bounded, the left side of (3.2) viewed as a derivative of a bounded function, is uniformly bounded in W 1, (Q T ). Consequently, ε x η(u ε, v) m ε {bounded of W 1, (Q T )}.

6 6 Wladimir Neves Moreover, m ε has total variation uniformly bounded over any compact set contained in Q T, that is, m ε M (Q T ). Hence applying Theorem 6 (Compactness for Measures) in Evans [3], we have {m ε } is pre-compact in W 1,q (Q T ), (1 q < n + 2 n + 1 ). Now, for any compact set D Q T and v ( R 0, R 0 ), ε x η(u ε, v) is a continuous linear functional in W 1,2 (Q T ). Indeed, for any function φ W 1,2 0 (Q T ), we have < ε x η(u ε, v), φ > = Q T ε x η(u ε, v)φ dx dt dv C ε( R 0 R 0 Q T ε x u ε 2 ) 1/2 ( R 0 R 0 Q T x φ 2 ) 1/2 C (3.3) ε x φ L 2, where C does not depend on ε. Taking the supremum in (3.3) with respect to the set W := {φ W 1,2 0 : φ W 1,2 1}, we obtain 0 ε x η(u ε, v) W 1,2 C ε sup φ. W Therefore passing to the limit as ε 0 + in the above inequality, the family {ε x η(u ε, v)} converges to zero in W 1,2, which implies that {ε x η(u ε, v)} is pre-compact in W 1,2 (Q T ). Now, W 1,2 (Q T ) has continuous imbedding in W 1,q (Q T ) for any 1 q < n+2 n+1, it follows that ε x η(u ε, v) m ε {bounded of W 1, (Q T )} {pre-compact of W 1,q (Q T )}, and applying Theorem 4.2 in Frid [7], we have Then, we obtain ε x η(u ε, v) m ε {pre-compact of W 1,2 (Q T )}. m ε {pre-compact of W 1,2 (Q T )}. 3. At this moment, we alize our problem in the following way. Given any compact set D Q T, we choose a smooth function ζ, such that ζ has compact support, ζ 1 on D, 0 ζ 1. Then, multiplying (2.3) by ζ and denoting fε (t, x, v) = ζ(t, x) f ε (t, x, v), we have t fε + a(v) x fε ε x fε = v (ζ m ε ) + f ε ζ t + f ε a(v) x ζ + 2ε x f ε x ζ. We would like to write the right side of the above equation, as a derivative with respect to v of a Radon measure in a compact set of W 1,2 (Q T ). Define m ε := ζ m ε + v f ε [ζ t + a(s) x ζ] ds + 2ε v x f ε x ζ ds. (3.4)

7 Scalar Viscous Conservation Laws 7 For any function φ W 1,2 0 (Q T ), we have < m ε, φ > < m ε, ζφ > + +2ε R0 Q T Q T R0 R 0 f ε φ a(s) t,x ζ ds dx dt dv R 0 f ε [ x ζφ + x ζ x φ ds dx dt dv C φ W 1,2 0 where we have used that f ε is uniformly bounded in L (Q T ), the compact support of ζ, and m ε W 1,2 (Q T ). So m ε is a continuous linear functional in W 1,2 (Q T ). Moreover, passing to a subsequence, we observe that < m ε, φ > converges in. Then, we have m ε { pre-compact of W 1,2 (Q T )}, and f ε satisfies t fε + a(v) x fε ε x fε = v m ε em D (Q T ). (3.5) For simplicity we maintain the notation f ε, m ε for f ε and m ε respectively. Now, the support of m ε is a fixed compact set contained in Q T. Consequently, we can write, see Lions, Pertame and Tadmor [8] v m ε (t, x, v) = (I 2 v)(i t,x ) 1/2 g ε, with g ε belonging to a pre-compact set of L 2 (Q T ). Then f ε satisfies t f ε + a(v) x f ε ε x f ε = (I 2 v)(i t,x ) 1/2 g ε. (3.6) 4. Let {ε l } l=1 be an arbitrary sequence which converges to zero as l. Let gε l be a subsequence of g ε, such that g ε l ḡ in L 2 (Q T ) when l and, passing to a subsequence if necessary, we have f ε l f in L (Q T ) when l. Clearly, f and ḡ satisfy (3.6). Moreover, we observe that f ε l, f have compact support in Q T ( R 0, R 0 ), so we have f, f ε l L 2 (Q T ). Now, we shall prove f ε l (t, x, v) dv f(t, x, v) dv in L 2 (Q T ). (3.7) It is enough to show that, for any function ψ C0 ( R 1, R 1 ) with R 1 > R 0 ˆfε l (τ, k, v)ψ(v) dv ˆ f(τ, k, v)ψ(v) dv in L 2 ( n ), (3.8) where ˆf ε l (τ, k, v) and ˆ f(τ, k, v) are the Fourier Transform of f ε l (t, x, v) and f(t, x, v) respectively. Indeed, by the Plancherel Identity, we obtain (3.7) taking ψ 1 over ( R 0, R 0 ) in (3.8). 5. Let f(t, x, v), g(t, x, v) L 2 ( n ) satisfying (3.6), and δ > 0 to be choose,

8 8 Wladimir Neves a posteriori. We take ζ C0 (), ζ 1 in ( 1, 1), ζ 0 in \ ( 2, 2) and following DiPerna & Lions [2], we write ˆf(τ, k, v) ψ(v) dv = I1 + I 2 ; (3.9) I 1 = τ + a(v) k ˆfψ(v)ζ( ) dv, I 2 = δ τ + a(v) k ˆfψ(v)[1 ζ( )] dv. δ For (τ, k) n with τ 2 + k 2 = 1, we define the following distribution function µ(s) : [0, ) [0, ], µ(s) µ τ,k (s) := L 1 ({v ( R 1, R 1 ); τ + a(v) k s}). Then, with τ 2 + k 2 > 0 we have I 1 = ˆf ψ(v) ζ( τ+a(v) k ) dv δ C ( v R 1 ˆf 2 ) 1/2 [µ( )] 1/2. (τ 2 + k 2 ) 1/2 δ (3.10) Now, we estimate I 2. Taking the Fourier Transform in (3.6) with respect to (t, x), we get ˆf(τ, k, v) = (1 + τ 2 + k 2 ) 1/2 ε k 2 + i(τ + a(v) k) (I 2 v)ĝ(τ, k, v). Then, we have ( ) 1/2 I 2 C (1 + τ 2 + k 2 ) 1/2 v R 1 ĝ 2 dv ( v R 1 1 { τ+a(v) k δ} { 1+ k 2 / δ 2 + k 4 / δ 4 τ+a(v) k 2 + k 2 + k 4 / δ 2 τ+a(v) k 4 + Now, observing that R1 R 1 1 { τ+a(v) k δ} τ + a(v) k p dv = 1 (τ 2 + k 2 ) p/2 } k 4 τ+a(v) k dv) 1/2. 6 δ (τ 2 + k 2 ) 1/2 see Folland [6] (Distribution Functions), we obtain from (3.11) 1 s p dµ(s) C δ p, (3.11) I 2 C (1 + τ 2 + k 2 ) 1/2 ( v <R 1 ĝ 2 dv) 1/2 ( δ 1 + k δ 2 + k 2 δ 3 ). (3.12) 6. Finally we prove (3.8), that is, (ˆ f ˆfε l ) ψ(v) dv 2 dτ dk X n l 0. Since ˆ f,ˆfε l L 2 ( n ), given δ > 0, there exist γ > 0, such that (τ 2 + k 2 ) 1/2 <γ v <R 1 (ˆ f ˆfε l ) 2 dv dτ dk < δ. (3.13)

9 Scalar Viscous Conservation Laws 9 Fixed δ > 0, we prove that (τ 2 + k 2 ) 1/2 γ (ˆ f ˆfε l ) ψ(v) dv 2 dτ dk 0, (3.14) l where we utilize (3.9), (3.10) and (3.12) with f and g replaced by f f ε l and ḡ g ε l respectively. Indeed, for (τ 2 + k 2 ) 1/2 γ, we choose δ = δ (τ 2 + k 2 ) 1/2, with δ > 0 a small number. Then by (3.10) and (3.12), we have ( 1/2 I 1 C [µ τ,k (δ )] 1/2 v <R 1 ˆ f ˆfε l dv) 2, ( ) 1/2 I 2 C h(τ, k) [(δ ) 1 + (δ ) 2 + (δ ) 3 ] ˆḡ ĝ ε l 2 dv v <R1, where h(τ, k) L ({τ 2 + k 2 γ}). Consequently, (ˆ f ˆfε l ) ψ(v) dv 2 dτ dk (τ 2 + k 2 ) 1/2 γ C 1 [µ τ,k (δ )] ˆ f ˆfε l 2 dv dτ dk (τ 2 + k 2 ) 1/2 γ v <R 1 + C 2 sup h(τ, k) 2 ˆḡ ĝ ε l 2 dv dτ dk. (τ 2 + k 2 ) 1/2 (τ 2 + k 2 ) 1/2 γ v <R 1 The Plancherel identity implies that ĝ ε l ˆḡ in L 2 ( n ), so the second term in the right side of the above inequality goes to zero. For the former, we have the following. Let ϖ : S n [0, 1] [0, ], ϖ(τ, k, s) := µ τ,k (s). It is not difficult to show that ϖ is uniformly continuous in S n [0, 1]. Indeed, since S n [0, 1] is a compact set, it is enough to prove that ϖ is continuous. Let (τ l, k l, s l ) S n [0, 1] be a convergent sequence, such that (τ l, k l, s l ) (τ ω, k ω, s ω ) S n [0, 1]. l Then, applying the Dominated Convergence Theorem, we get µ τ l,k l (s l ) µ τ ω,k ω (s ω ) R1 R 1 1 { τl +a(v) k l s l } 1 { τω +a(v) k ω s ω } dv l 0, which implies that ϖ is continuous in S n [0, 1]. Moreover, we have ϖ(τ, k, 0) = L 1 ({v ( R 1, R 1 ); τ + k a(v) = 0}) = 0, for all (τ, k) S n. Hence we obtain that [µ τ,k (δ )] ˆ f ˆfε l 2 dv dτ dk (τ 2 + k 2 ) 1/2 γ v <R 1 sup µ τ,k (δ ) ˆ f ˆfε l 2 dv dτ dk (τ,k) S n (τ 2 + k 2 ) 1/2 γ v <R 1 and, since δ can be take arbitrary small, the final result follows.

10 10 Wladimir Neves Acknowledgments This research was partially supported by FUJB, under the grant Resumo.Utilizamos a teoria cinética para estudar a compacidade da família {u ε } ε>0, soluções do problema de valor inicial e de contorno para leis de conservação escalares viscosas u ε t + div xf(u ε ) = ε xu ε em domínios não cilíndricos. Considerando f em C 3 e satisfazendo a condição de ser não degenerada, provamos que u ε é compacta em L 1. References [1] Chen, G.-Q. and Frid, H., Large-time behavior of Entropy Solutions in L for Multidimensional Scalar Conservation Laws, Advances in nonlinear partial differential equations and related areas. World Sci. Pub., River Edge, NJ.(1998), [2] DiPerna, R. and Lions, P. L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), [3] Evans, L. C., Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics No. 74. AMS: Providence, [4] Evans, L. C. and Gariepy, R. F., Lecture Notes on Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, Florida, [5] Federer, H., Geometric Measure Theory, Springer-Verlag, New York, [6] Folland, G. B., Real Analysis: modern techniques and their applications, John Willey & Sons, Inc, [7] Frid, H., Compacidade Compensada Aplicada as Leis de Conservação, 19 Colóquio Brasileiro de Matemática, IMPA. [8] Lions, P. L., Perthame, B., and Tadmor, E., A Kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), [9] Neves, W., Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains, J. Differential Equations 192 (2003), [10] Perthame, B., Kinetic Formulation of Conservation Laws, Oxford Lecture Series in Mathematics and Its Applications, [11] Schwartz, L., Théorie des Distributions (2 volumes), Actualites Scientifiques et Industrielles 1091, 1122, Herman, Paris, [12] Serre, D., Systems of Conservation Laws, Vols. 1 2, Cambridge: Cambridge University Press, 1999.