Asymptotic Behavior of Smooth Solutions for Dissipative Hyperbolic - PowerPoint PPT Presentation

Asymptotic Behavior of Smooth Solutions for Dissipative Hyperbolic Systems with a Convex Entropy Stefano Bianchini - IAC (CNR) ROMA Bernard Hanouzet - Math´ ematiques Appliqu´ ees de Bordeaux Roberto Natalini - IAC (CNR) ROMA

ć ą ¡ ű ţ ű ţ Hyperbolic systems of balance laws Consider a system of balance laws with k conserved quantities, ∂ t u + ∂ x F 1 ( w ) = 0 (1) ∂ t v + ∂ x F 2 ( w ) = q ( w ) with w = ( u, v ) ∈ Ω ⊂ R k × R n − k , and assume that there exists a strictly convex function E = E ( w ) and a related entropy-flux F = F ( w ), s.t. (for smooth solutions): ∂ t E ( w ) + ∂ x F ( w ) = G ( w ) , (2) where F ′ 0 F ′ = E ′ F ′ ( w ) = E ′ 1 G = E ′ G ( w ) = E ′ , . F ′ q ( w ) 2 Equilibrium points: ¯ w s.t. G ( ¯ w ) = 0. Set γ = { w ∈ Ω; G ( w ) = 0 } . Definition. The system (1) is entropy dissipative, if for every ¯ w ∈ γ and w ∈ Ω , E ′ ( w ) − E ′ ( ¯ R ( w, ¯ w ) := w ) · G ( w ) ≤ 0 .

ű ţ Set W = ( U, V ) = E ′ ( w ), Φ( W ) := ( E ′ ) − 1 ( W ), and rewrite (1) in the symmetric form A 0 ( W ) ∂ t W + A 1 ( W ) ∂ x W = G (Φ( W )) (3) with A 0 ( W ) := Φ ′ ( W ) symmetric, positive definite and A 1 ( W ) := F ′ (Φ( W ))Φ ′ ( W ) symmetric. The system (3) is strictly entropy dissipative , if there exists a positive definite W ) ∈ M ( n − k ) × ( n − k ) such that matrix B = B ( W, ¯ Q ( W ) := q (Φ( W )) = − D ( W, ¯ W )( V − ¯ V ) , (4) for every W ∈ E ′ (Ω) and ¯ W = ( ¯ U, ¯ V ) ∈ Γ := E ′ ( γ ) = { W ∈ E ′ (Ω); G (Φ( W )) = 0 } . In the following we just consider ¯ W = 0 and systems like: 0 A 0 ( W ) ∂ t W + A 1 ( W ) ∂ x W = − , (5) D ( W ) V with D positive definite.

ć ţ ą ű Kawashima condition. Consider our original system ∂ t w + F ′ ( w ) ∂ x w = G ( w ) . (6) Condition K. Any eigenvector of F ′ (0) is not in the null space of G ′ (0), which can be rewritten in entropy framework as U [ λA 0 (0) + A 1 (0)] � = 0 ( K ) 0 Theorem 1. (Hanouzet-Natalini) Assume that system (5) is strictly entropy dissipa- tive and condition (K) is satisfied. Then there exists δ > 0 such that, if � W 0 � 2 ≤ δ , there is a unique global solution W = ( U, V ) of (5), which verifies W ∈ C 0 ([0 , ∞ ); H 2 ( R )) ∩ C 1 ([0 , ∞ ); H 1 ( R )) , and Z + ∞ � W ( t ) � 2 � ∂ x U ( τ ) � 2 1 + � V ( τ ) � 2 dτ ≤ C ( δ ) � W 0 � 2 sup 2 + 2 , (7) 2 0 ≤ t< + ∞ 0 where C ( δ ) is a positive constant. In multiD the estimate is in H s , with s sufficiently large (Yong).

ÿ ÿ ů ÿ ů ÿ ů ÿ ů ů ÿ ů The linearized problem. The system of balance law (1) becomes A 11 A 12 0 0 ∂ t w + ∂ x w = − w, (8) A 21 A 22 D 1 D 2 (H1) ∃ A 0 symmetric positive such that AA 0 is symmetric and A 0 , 11 A 0 , 12 0 0 A 0 = , BA 0 = − , A 0 , 21 A 0 , 22 0 D with D ∈ R ( n − k ) × ( n − k ) positive definite; (H2) any eigenvector of A is not in the null space of B . Consider the projectors Q 0 = R 0 L 0 on the null space of B , and its complementary projector Q − = I − Q 0 = R − L − , to which it corresponds the decomposition ( A 0 , 11 ) − 1 / 2 0 w = A 0 w c + w nc , (9) (( A − 1 0 ) 22 ) − 1 / 2 0 h i h 0 ) 22 ) − 1 / 2 i ( A 0 , 11 ) − 1 / 2 (( A − 1 w c = u, w nc = A 0 u. (10) 0 0

ů ű ÿ ů ű ţ ÿ ů ű ů ¡ The system (8) takes now the form ˜ ˜ ÿ ţ ÿ ţ 0 0 w c A 11 A 12 w c w c + = , (11) ˜ ˜ ˜ w nc w nc 0 D w nc A 21 A 22 t x where ˜ A is symmetric and ˜ D is strictly negative, D . ˜ = L − ˜ BR − = (( A − 1 0 ) 22 ) − 1 D (( A − 1 0 ) 22 ) − 1 . We want to study the Green kernel Γ( t, x ) of (11), ∂ t Γ + ˜ ˜ 0 0 A∂ x Γ = B Γ ˜ B = , ˜ Γ(0 , x ) = δ ( x ) I 0 D by means of Fourier transform ˆ Γ( t, ξ ) and perturbation analysis of the characteristic function E ( z ) = ˜ B − zA. We will consider the Green kernel as composed of 4 parts, Γ 00 ( t, x ) Γ 0 − ( t, x ) Γ( t, x ) = . Γ − 0 ( t, x ) Γ −− ( t, x )

For ξ small (large space scale), the reduction of E ( z ) on the eigenspace of the 0 eigenvalue of ˜ B is A 11 − z 2 ˜ D − 1 ˜ − z ˜ A 12 ˜ A 21 + O ( z 3 ) , and one has to consider the decomposition D − 1 ˜ ˜ l j ˜ A 12 ˜ X X A 11 = ℓ j r j l j , A 21 r j = ( c jk I + d jk ) p jk , j k with d jk nilpotent matrix. Let us denote by g jk ( t, x ) the heat kernel of g t + ℓ j g x = ( c jk I + d jk ) g xx . For ξ large (small space scale), E ( z ) = z ( ˜ A + ˜ B/z ), one has to consider the decomposition ˜ X L j ˜ X A = λ j R j L j , BR j = ( b jk I + e jk ) q jk , j k and let h jk ( t, x ) be Green kernel of the transport system h t + λ j h x = ( b jk I + e jk ) h.

ů ł ľ ľ ÿ ł Define the matrix valued functions dx r j g jk ( t, x ) p jk l j ˜ A 12 ˜ − d D − 1 2 r j g jk ( t, x ) p jk l j 3 X K ( t, x ) = 4 5 D − 1 ˜ D − 1 ˜ d 2 dx ˜ dx 2 ˜ A 21 r j g jk ( t, x ) p jk l j ˜ A 21 ˜ − d D − 1 A 21 r j g jk ( t, x ) p jk l j jk X K ( t, x ) = R j ( h jk ( t, x ) q jk ) L j . jk Theorem. The Green kernel for (11) is λt ≤ x ≤ ¯ λt ≤ x ≤ ¯ Γ( t, x ) = K ( t, x ) χ λt, t ≥ 1 + K ( t, x ) + R ( t, x ) χ λt , (12) where λ , ¯ λ are the minimal and maximal eigenvalue of ˜ A and the rest R ( t, x ) can be written as e − ( x − ℓ j t ) 2 /ct O (1)(1 + t ) − 1 / 2 O (1) X R ( t, x ) = O (1)(1 + t ) − 1 / 2 O (1)(1 + t ) − 1 1 + t j for some constant c .

ć ą Differences with the previous result by Y. Zeng (1999): 1. finite propagation speed (hyperbolic domain); 2. Structure of the diffusive part (operators R 0 and L 0 ); 3. BA 0 not symmetric ⇔ ˜ D not symmetric (as in Hanouzet-Natalini (2002), Yong (2002)). From a technical point of view, when we study the function ˆ ( ˜ B − z ˜ G ( t, ξ ) = exp( E ( z ) t ) = exp A ) t , and we compute its inverse Fourier transform, the differences w.r.t. Y. Zeng are: • a carefully analysis of the families of eigenvalues whose projectors do not blow up near the exceptional points z = 0, z = ∞ ; • when estimating e E ( z ) t , one has to deal always with matrices; • the path of integration in the complex plane depends now on the viscosity coefficients c jk , which is a complex number.

ć ą ţ ţ ć ű ű ţ ű ć ą ą Asymptotic behavior Consider now the original problem 0 w t + F ( w ) x = G ( w ) = , w ( x, 0) = w 0 (13) q ( w ) We have w t + F ′ (0) w x − G ′ (0) w = F ′ (0) w − F ( w ) G ′ (0) w − G ( w ) x − Then we can write the solution as Z t ş ą ć ť F ′ (0) w − F ( w ) G ′ (0) w − G ( w ) w = Γ( t ) ∗ w 0 + Γ( t − τ ) ∗ x − dτ. 0 Since for any vector vector (0 , V ) ∈ R k × R n − k one has for the principal part K of the kernel Γ − r j g jk ( t, x ) p jk l j ˜ A 12 ˜ D − 1 0 d X K ( t, x ) = , D − 1 ˜ dx ˜ A 21 r j g jk ( t, x ) p jk l j ˜ A 21 ˜ d D − 1 V dx jk also the second term in the convolution contains an x derivative, so that one may use standard L 2 estimates.

ľ ľ Theorem. Let u ( t ) be the solution to the entropy strictly dissipative system (13) , and let w c ( t ) = L 0 w ( t ) , w nc ( t ) = L − w ( t ) . Then, if � u (0) � H s is bounded and small for s sufficiently large, the following decay estimates holds: for all β , n 1 , t − 1 / 2(1 − 1 /p ) − β/ 2 o � ∂ β x w c ( t ) � L p ≤ C min max � u (0) � L 1 , � u (0) � H s ł , (14) n 1 , t − 1 / 2(1 − 1 /p ) − 1 / 2 − β/ 2 o � ∂ β x w nc ( t ) � L p ≤ C min max � u (0) � L 1 , � u (0) � H s ł , (15) with p ∈ [1 , + ∞ ] . 2 πt e − x 2 / 4 t , 1 Remark. These decay estimates correspond to the decay of the heat kernel √ and in particular the solution to the linearized problem w t + ˜ Aw x = ˜ Bw satisfies (14), (15). As a consequence these estimates cannot be improved. Remark. Observe moreover that the non conservative variables w nc decays as a deriva- tive of w c .

ű ą ţ ş ť Chapman-Enskog expansion Consider now the Chapman-Enskog expansion 0 A 0 ( W ) ∂ t W + A 1 ( W ) ∂ x W = − , W = ( U, V ) D ( W ) V V ∼ h ( U, U x ) := − D − 1 ş ( A 1 ) 21 − ( A 0 ) 21 ( A 0 ) − 1 11 ( A 1 ) 11 U x In the original coordinates, equilibrium at v = h ( u ) and ć ť u, h ( u ) − D − 1 ( u, h ( u )) u t + F 1 F 2 ( u, h ( u )) x − Dh ( u ) F 1 ( u, h ( u )) x x = 0 (16) The linearized form of (16) is D − 1 ˜ u t + ˜ A 11 u x − ˜ A 12 ˜ A 21 u xx = 0 , so that its Green kernel G is Γ( t ) = K 00 ( t ) + ˜ ˜ K ( t ) + ˜ X R ( t ) , K 00 ( t, x ) = r j g jk ( t, x ) p jk l j . jk

ľ Since the principal part of the linear Green kernel is the same (up to the finite speed of propagation), one can prove Theorem. If w ( t ) is the solution to the parabolic system (16) , then for all κ ∈ [0 , 1 / 2) n 1 , t − m/ 2(1 − 1 /p ) − κ − β/ 2 o � D β ( w c ( t ) − w ( t )) � L p ≤ C min max � u (0) � L 1 , � u � H s ł , if the initial data is sufficiently small, depending on κ , and tending to 0 as κ → 1 / 2 . Remark. At the linear level one gains exactly t − 1 / 2 (one derivative), but in dimension 1 the quadratic parts of F , G matter and this is way we can only prove the decay for all k ∈ [0 , 1 / 2).

A Glimm Functional for Relaxation Stefano Bianchini - IAC (CNR) ROMA