Decay rates for partially dissipative hyperbolic systems

Transcription

1 Outline Decay rates for partially dissipative hyperbolic systems Basque Center for Applied Mathematics Bilbao, Basque Country, Spain Numerical Methods for Hyperbolic Problem, Chevaleret, November 2009 Joint work with Karine BEAUCHARD, ENS-Cachan, France

2 Outline Outline

3 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

4 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

5 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

6 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

7 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

8 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

9 Outline Outline 1 Introduction 2 Preliminaries in control theory 3 Systems of balance laws 4 Partially dissipative linear hyperbolic systems: (SK) and beyond 5 Back to nonlinear 6 Connections with hypoellipticity & hypocoercivity 7 Conclusions

10 Introduction Partially dissipative linear hyperbolic systems may develop a complex asymptotic behavior as t depending on the space dimension, the dimension of the system, and the interaction between the free dynamics and the damping operator. The understanding of this issue is relevant, in particular, for analyzing global existence of solutions for nonlinear problems, near constant equilibria. The issue turns out to be closely linked to the classical Kalman rank condition for the control of finite-dimensional linear systems, the (SK) condition and other notions such as hypoellipticity and hypocoercivity of partially diffusive PDEs.

11 Introduction Partially dissipative linear hyperbolic systems may develop a complex asymptotic behavior as t depending on the space dimension, the dimension of the system, and the interaction between the free dynamics and the damping operator. The understanding of this issue is relevant, in particular, for analyzing global existence of solutions for nonlinear problems, near constant equilibria. The issue turns out to be closely linked to the classical Kalman rank condition for the control of finite-dimensional linear systems, the (SK) condition and other notions such as hypoellipticity and hypocoercivity of partially diffusive PDEs.

12 Introduction Partially dissipative linear hyperbolic systems may develop a complex asymptotic behavior as t depending on the space dimension, the dimension of the system, and the interaction between the free dynamics and the damping operator. The understanding of this issue is relevant, in particular, for analyzing global existence of solutions for nonlinear problems, near constant equilibria. The issue turns out to be closely linked to the classical Kalman rank condition for the control of finite-dimensional linear systems, the (SK) condition and other notions such as hypoellipticity and hypocoercivity of partially diffusive PDEs.

13 The Kalman rank condition Controllability of finite dimensional linear systems Let n, m N and T > 0. Consider the following finite dimensional system: { x (t) = Ax(t)+Bu(t), t (0, T ), x(0) = x 0. Here A is a real n n matrix, B is a real n m matrix, x : [0, T ] R n represents the state and u : [0, T ] R m the control. System (1) is controllable in time T > 0 if given any initial and final one x 0, x 1 R n there exists u L 2 (0, T, R m ) such that the solution of (1) satisfies x(t ) = x 1. Obviously, in practice m n. Of particular interest is the case where m is as small as possible. (1)

14 The control property does not only depend on the dimensions m and n but on how the matrices A and B interact x (t) = Ax(t) + Bu(t) Theorem (Kalman, 1958) The system (A, B) is controllable in some time T if and only if rank [B, AB,, A n 1 B] = n. (2) Consequently, if system (1) is controllable in some time T > 0 it is controllable in any time. Note that, the so-called controllability matrix [B, AB,, A n 1 B] is of dimension n nm. Thus, in the limit case where m = 1 (one single control), it is a n n matrix. In this case the Kalman rank condition requires it to be invertible.

15 Proof of Theorem 1: Suppose that rank([b, AB,, A n 1 B]) < n. From Cayley-Hamilton Theorem we deduce that rank([b, AB,, A n 1 B,...]) < n. Thus, the rank of e At B is confined in a vector space of dimension < n for all t > 0. From the variation of constants formula, the solution x of (1) satisfies t x(t) = e At x 0 + e A(t s) Bu(s)ds. 0 But t 0 ea(t s) Bu(s)ds is confined to a subspace of dimension < n. Consequently, there are non trivial directions v R n such that v, x(t ) = v, e AT x 0 + T 0 v, e A(T s) Bu(s) ds = v, e AT x 0.

16 Stabilization of finite dimensional linear systems The controls we have obtained so far are the so called open loop controls. In practice, it is interesting to get closed loop or feedback controls, so that its value is related in real time with the state itself. Assume, to fix ideas, that A is a skew-adjoint matrix, i. e. A = A. In this case, < Ax, x >= 0. Consider the system { x = Ax+Bu x(0) = x 0 (3). When u 0, the energy of the solution of (3) is conserved. Indeed, by multiplying (3) by x, if u 0, one obtains Hence, d dt x(t) 2 = 0. x(t) = x 0, t 0.

17 We then look for a matrix L such that the solution of system (3) with the feedback control law u(t) = Lx(t) has a uniform exponential decay, i.e. there exist c > 0 and ω > 0 such that x(t) ce ωt x 0 for any solution. In other words, we are looking for matrices L such that the solution of the system x = (A + BL)x = Dx has an uniform exponential decay rate.

18 Theorem If A is skew-adjoint and the pair (A, B) is controllable then L = B stabilizes the system, i.e. the solution of { x = Ax BB x x(0) = x 0 (4) has an uniform exponential decay. Proof: With L = B we obtain that 1 d 2 dt x(t) 2 = < BB x(t), x(t) >= B x(t) 2 0. Hence, the norm of the solution decreases in time. Then, use the same ideas as before in the way you prefer to conclude.

19 An example: The harmonic oscillator Consider the damped harmonic oscillator: mx + Rx+kx = 0, (5) where m, k and R are positive constants. It is easy to see that the solutions of this equation have an exponential decay property. Indeed, it is sufficient to remark that the two characteristic roots have negative real part. Indeed, and therefore mr 2 + R + kr = 0 r ± = k ± k 2 4mR 2m Re r ± = { k 2m k 2m ± k 2 4m R 2m if k 2 4mR if k 2 4mR.

20 We observe here the classical overdamping phenomenon. Contradicting a first intuition it is not true that the decay rate increases when the value of the damping parameter k increases.

21 Systems of balance laws We consider nonlinear hyperbolic systems of the form w m t + j=1 F j (w) x j = Q(w), x R m, t > 0, w : R R m R n (t, x) w(t, x) arising in so many applications: fluid mechanics, gas dynamics, traffic flow, ralaxation,... Local (in time) smooth solutions are known to exist (C. Dafermos, L. Hsiao-T.-P. Liu, A. Majda, D. Serre,...) But possible singularities (i.e. shock waves) may arise in finite time. The nonlinear term Q may play the role of a partial dissipation and help to the existence of global smooth solutions.

22 Global smooth solutions in a neighborhood of a constant equilibrium: Q(w ) = 0. THE METHOD = LINEARIZATION + FIXED POINT. If the linearized dynamics exhibits solutions that decay as t, a perturbation argument may hopefully allow showing that, locally around the constant equilibrium, solutions are global and decay as well. One has to distinguish: Total dissipation : Q(w) = Bw, ( B > ) Partial dissipation: Q(w) = w. 0 D Y. Shizuta & S. Kawashima (85), Y. Zeng (99), W. A. Yong (04), S. Bianchini, B. Hanouzet & R. Natalini (03, m = 1),... Example : Isentropic Euler equations with damping u t v x = 0, v t + f (u) x = v

23 Partially dissipative linear hyperbolic systems A 1,..., A m symmetric w m t + w A j = Bw, x R m, w R n (6) x j j=1 B = ( D ) n1 n 2 X t DX > 0 X R n 2 {0} Goal: Understand the asymptotic behavior as t. Apply Fourier transform: ŵ m t = ( B ia(ξ))ŵ where A(ξ) := ξ j A j Lack coercivity : [B + ia(ξ)]x, X = BX, X = DX 2, X 2 c X 2 But possible decay depending on ξ: exp[( B ia(ξ))t] Ce λ(ξ)t j=1

24 PARTIALLY DISSIPATIVE LINEAR HYPERBOLIC SYSTEM m-parameter (ξ) FAMILY OF FINITE-DIMENSIONAL PARTIALLY DISSIPATIVE n-dimensional SYSTEMS. The asymptotic behavior of solutions is determined by the behavior of the function ξ λ(ξ) giving the decay rate as a function of ξ. Example: The dissipative wave equation u tt u xx + u t = 0. u t = v x u; v t = u x. Solutions may be decomposed as A high frequency component tending to zero exponentially fast as t 0; A low frequency component with the same decay rate as the heat kernel t 1/2 decay in L for L 1 data. This corresponds to a function λ(ξ) min( ξ 2, 1).

25 (SK) for partially dissipative linear hyperbolic systems The pioneering and well-known result by Shizuta-Kawashima (85) may be viewed as a generalization of the example above on the dissipative wave equation. Under the condition (SK) below (SK) : ξ R m, Ker(B) {eigenvectors of A(ξ)} = {0} exp[( B ia(ξ))t] Ce λ(ξ)t, λ(ξ) = c min(1, ξ 2 ) Their proof is based on a (long) linear algebra computation. Consequences : (1) w 0 L 1 L 2 (R m, R n ), w = w l + w w h w h (t) L 2 (R m,r n ) Ce γt 0 L 2 (R m,r n ) w l (t) L (R m,r n ) Ct m 2 w 0 L 1 (R m,r n ) (2) Global smooth solutions for the NL system for initial data close to a constant equilibrium.

26 Our contributions (K. Beauchard & E. Z., ARMA, to appear) 1st step : (SK) and Kalman rank condition 2nd step : Measure of the decay rate for B + ia(ξ) 3rd step : Classification of the asymptotic behavior for linear hyperbolic systems (with/without (SK)) 4th step : Nonlinear systems of balance laws : global smooth solutions without (SK)

27 (SK) and the Kalman rank condition

28 (SK) and Kalman rank conditions Lemma The (SK) condition is equivalent to imposing the Kalman rank condition to the pairs (A(ξ), B) for all values of ξ R m. A symmetric B = ( D (SK) : Ker(B) {eigenvectors of A} = {0} ) n1 n 2 X t DX > 0 X R n 2 {0} The pair (A, B) satisfies the Kalman rank condition Be At X 0 for all t > 0 = X 0 n 1 k=0 BA k X 2 is a norm on R n (the Kalman quadratic form)

29 The (SK) condition requires the conditions above to be satisfied for all ξ R m. In 1 d (m = 1) the (SK) condition is sharp. Whenever it fails, travelling wave solutions with L 2 -profiles exist, thus making the decay of solutions impossible. This is so because, in 1 d, the Kalman condition holds for all ξ R if and only if the pair (A, B) sastifies the Kalman condition. This is not true in the muti-dimensional case since A(ξ) = A(ξ) := m ξ j A j depends on ξ in a non-trivial way. j=1 In view of the analysis above it is more natural to analyse the positive definiteness of the quadratic form n 1 BA(ξ) k X 2, as k=0 function of ξ. This will illustrate the existence of many other scenarios that (SK) excludes, except in one space dimension (m = 1). In the multi-dimensional case (SK) is a sufficient condition for decay, but is far from being necessary.

30 Decay rate for B + ia(ξ) as a function of ξ

31 A measure of the decay rate as a function of ξ ( ) A 1,..., A m 0 0 n1 B = A(ξ) := m ξ symmetric 0 D n j A j 2 ξ = ρω R m ρ > 0 ω S m 1 (m k ) well chosen Theorem n 1 N,ɛ (ω) := min{ ɛ m k BA(ω) k x 2 ; x S n 1 }. k=0 ɛ > 0, c > 0 such that ɛ (0, ɛ ), exp[( B iρa(ω))t] 2e cn,ɛ(ω)min{1,ρ2 }t. Remark : (SK) N,ɛ (ω) N,ɛ > 0, ω S m 1. In general, N,ɛ (ω) may vanish for some values of ω S m 1, in which case the decomposition of solutions and its asymptotic form is more complex. j=1

32 Our proof: Is based on energy arguments and the construction of a suitable Lyapunov functional that allows exhibiting the exponential decay rate. In that sense it is similar to the techniques employed for proving decay rates for dissipative wave equations, and the works by C. Villani et al. on the decay for kinetic equations and hypocoercivity. Yields the result by Shizuta-Kawashima in a simpler way, but shows that it only covers one of the many possible behaviors one may encounter for m 2. Provides quantitative estimates on the decay rate as a function of ξ that can be used for a better understanding of the nonlinear problem.

33 An example: the dissipative wave equation. u tt u xx + u t = 0. w tt + ξ 2 w + w t = 0. e ξ (t) = 1 2 [ w t 2 + ξ 2 w 2 ]. de ξ (t) = w t 2. dt The exponential decay rate does not come directly out of this. But it is easy to check that is, for ε small enough, such that f ξ (t) = e ξ (t) + εww t, and, on the other hand, df ξ (t) dt f ξ e ξ c(ξ, ε)f ξ (t).

34 Indeed, df ξ (t) dt = w t 2 + ε w t 2 + εww tt = (1 ε) w t 2 εξ 2 w 2 εww t (1 ε ε 2ξ 2 ) w t 2 εξ2 2 w 2. Then, it suffices to take: ε = min ( ξ 2, 1 ). 4 Note that there is an extensive literature on the extensions of this result to nonlinear problems (M. Nakao, A. Haraux,... ).

35 Decay rate for B + ia(ξ) : ρ < 1 ω S m 1 : n 1 k=0 ẋ = ( B iρa(ω))x, ɛ m k BA(ω) k x 2 N,ɛ > 0, x S n 1. Goal : x(t) 2 x 0 e cn,ɛρ2 t Strategy : find L(x) x 2 such that dl dt cρ 2 N,ɛ L n 1 L(x) = x 2 + ρ ɛ m k Im ( A(ω) k BBA(ω) k 1 x, x ) k=1 dl dt = 2Re ( (B + iρa ω )x, x ) ρ n 1 k=1 ɛm k Im ( (A t ω) k B t BA k 1 ω (B + iρa ω )x, x ) ρ n 1 k=1 ɛm k Im ( (A t ω) k B t BAω k 1 x, (B + iρa ω )x ) 2C 1 Bx 2 ρ 2 n 1 k=1 ɛm k BA k ωx 2 +ρ n 1 k=1 ɛm k Bx [ BA k 1 ω BA k ωx + BA k ω BA k 1 ω x ] +ρ 2 n 1 k=1 ɛm k BA k 1 ω x BA k+1 ω x

36 Asymptotic behavior for linear hyperbolic systems (with or without (SK))

37 The set of degeneracy The minimum of Kalman s quadratic form N,ɛ (ω) := min{ n 1 k=0 ɛm k BA(ω) k x 2 ; x S n 1 } measures the decay rate for B + iρa(ω) N,ɛ (ω) > 0 Ker(B) {eigenvectors of A(ω)} = {0} (A(ω), B) satisfies the Kalman rank condition. The set of degeneracy : D(B + ia(ξ)) = {ξ R m ; rank[b BA(ξ)... BA(ξ) n 1 ] < n} is an algebraic submanifold either D = 0 N,ɛ > 0 a.e. strong L 2 stability or D = R m : non dissipated solutions Decomposition?

38 D = 0, n 1 = 1 (B is effective in all but one components) Theorem When n 1 = 1, D is a vector subspace of R m and N,ɛ (ω) c min{1, dist(ω, D) 2 }, ω S m 1. As a consequence we have the following new decomposition w = w h + w l + w new with wh w (t) L 2 (R m,r n ) Ce t 0 L 2 (R m,r n ) w l (t) L (R m,r n ) Ct m 2 w 0 L 1 (R m,r n ) w new (t) L (R m,r n ) Ct 1 2 w 0 Example: n = m = 2; D = {(ξ 1, ξ 2 ) : a 1 21 ξ 1 + a 2 21 ξ 2 = 0}.

39 Classification of the asymptotic behaviors w m t + w A j = Bw, x R m, w R n x j j=1 (SK) D L 2 stability decomposition m = 1 (SK) {0} yes e t + 1 t n no (SK) R no e t + 1 t +trav. waves n = 2 (SK) {0} yes m no (SK) hyperplane yes e t + 1 t e t + 1 t + 1 t no (SK) R m no e t + trav. wave n (SK) {0} yes e t + 1 t m/2 m no &n 1 = 1 vect. subsp. yes e t t m/2 t no (SK) submanifold yes open R m no open

40 Existence of global smooth solutions with (SK) (Yong, 04) w m t + j=1 ( F j (w) 0 0 = Bw B := x j 0 I n2 W e constant equilibrium with (SK) on its linearized system. ) n1 n 2 Theorem Let s [m/2] + 2 be an integer. There exists δ > 0 such that, w 0 W e + H s (R m, R n ) with H w 0 W e δ s there exists a unique global solution w C 0 ((0, + ) t, W e + H s ). Proof : Local existence + continuation argument. It uses the decay rate of the linearized system around W e.

41 Existence of global smooth solutions without (SK) inspired by Coron s return method The same result of global existence holds under the following assumptions: (H1) : A j := df j (w) symmetric (H2) : W p W in R n constant equilibria with N (W p ) > 0, N (W ) = 0 and N (W p ) >> W p W C(W p ) n 1 C(W p ) := sup{ ɛ m k A Wp (ω) k B t BA Wp (ω) k 1 ; ω S m 1 } k=1

42 Example w t + F (w) x = Bw B := ( ) F = (H1) F 1 w 2 = F 2 w 1 (H2) W = (0, 0) with F 1 w 2 (0, 0) = 0 no (SK) W p = (a p, 0) with F 1 w 2 (a p, 0) 0 (SK) We assume that M > 0 such that F 1 w 1 + F 2 w 2 M F 1 Then C(W p ) thus the theorem applies if N (W p ) F 1 w 2 (a p, 0) w 2. F 1 2 F 1 (a p, 0) >> a p (0, 0) >> 1 w 2 w 2 w 1 ( F1 F 2 )

43 Connections with hypoellipticity & hypocoercivity Hypoellipticity The fundamental solution is C away form the diagonal (...L. Hörmander, ) t u n a jk jk 2 u + j,k=1 n b jk x j k u = 0. j,k=1 Example: Kolmogorov equation: u t u xx xu y = 0. After applying Fourier transform: U t A(ξ, ξ)u n b jk j (ξ k U) = 0. j,k=1 Hypoellipticity fails iff A(e Bsξ, e Bsξ ) = 0 for all s > for some ξ 0.

44 Hypocoercivity (...C. Villani, 2006,...) L = A A + B, f t + Lf = 0 where B = B. Example: Similar models as above but in the context of kinetic equations: Fokker-Planck equation: f t v f + v x f V (x) v f v (vf ) = 0. Despite the lack of coercivity of L, under suitable assumptions on the commutators of A and B, one can build a modified energy or suitable Lyapunov functional (similar to our construction of the functional L) in which the time exponential decay can be proved.

45 Null control of the Kolmogorov equation: f f +v t x 2 f v 2 = u(t, x, v)1 ω(x, v), (x, v) R x R v, t (0, + ), [ ] (7) where ω = R x R v [a, b]. Equivalently, one may address the following observability inequality for the adjoint system: { g t v g x 2 g = 0, (x, v) R v 2 x R v, t (0, T ), (8) g(0, x, v) = g 0 (x, v), (x, v) R x R v, satisfies R x R v g(t, x, v) 2 dxdv C T 0 ω g(t, x, v) 2 dxdvdt.

46 Ideas of the proof: Fourier transform in v: { ˆf t (t, ξ, v) + iξvˆf (t, ξ, v) 2ˆf (t, ξ, v) = û(t, ξ, v)1 v 2 R [a,b] (v), ˆf (0, ξ, v) = ˆf 0 (ξ, v). (9) Decay: ˆf (t, ξ,.) ˆf t 0 (ξ,.) 3 /12 L 2 (R) L 2 (R) e ξ2, ξ R, t R +. Control depending on the parameter ξ with cost e C(T ) max{1, ξ }. (10)

47 Conclusion Control theoretical tools applied to analyze the decay rate for B + ia(ξ) yield: Goals: For linear hyperbolic systems : a (yet non complete) classification for the asymptotic behaviors, with and without (SK), for nonlinear systems of conservation laws : existence of global smooth solutions around some degenerate constant equilibria. To make the classification of linear systems complete; Significantly enlarge the class of nonlinear systems for which global existence holds.

48 Existence of global smooth solutions : proof w m t + A j (w) w = Bw x j j=1 Strategy : local existence + continuation argument 2 T w(t ) W e + 2 w 2 + C 1N (W p H s H s e ) 2 x w 2 w0 H W e H s 1 s 1st step : 2nd step : w(t ) 0 L 2 -estimate w(t ) W e 2 L H s -estimate 2 H s + 2 T 0 T 0 w 2 2 L 2 w0 2 L 2 w 2 2 H s w0 2 H s + CN s(t ) H N s (T ) := sup{ w(t) W e ; t (0, T )} s T 0 w 2 H s 1

49 3rd step : estimate on T 0 w 2 L 2 w m t + A j (We p ) w = Bw + h p x j j=1 N (W p e )= hypocoercivity for the linear system 2 w(t ) W e w 0 W e 2 + T H s 0 w N (W p H s e ) 2 w 2 x w + (N s(t ) + W e W p H s e 2 ) T 0 If N s (T ) + W e W p e 2 ) < N (W p e ) 2 then w(t ) W e H s < w0 W e H s. H s 1 2 H s 1 This always holds when H w 0 W e < 1 s 2 N (We p ) 2 W e We p < 1 2 N (We p )