Non-stationary Friedrichs systems

Transcription

1 Department of Mathematics, University of Osijek BCAM, Bilbao, November 2013 Joint work with Marko Erceg

2 1 Stationary Friedrichs systems Classical theory Abstract theory 2 3

3 Motivation Stationary Friedrichs systems Classical theory Abstract theory Introduced in: Goal: K. O. Friedrichs: Symmetric positive linear differential equations, Communications on Pure and Applied Mathematics 11 (1958), treating the equations of mixed type, such as the Tricomi equation: y 2 u x u y 2 = 0 ; unified treatment of equations and systems of different type.

4 Friedrichs system Stationary Friedrichs systems Classical theory Abstract theory Assumptions: d, r N, Ω R d open and bounded with Lipschitz boundary; A k W 1, (Ω; M r (R)), k 1..d, and C L (Ω; M r (R)) satisfying (F1) matrix functions A k are symmetric: A k = A k ; (F2) ( µ 0 > 0) C + C + d k A k 2µ 0 I (ae on Ω). k=1 Operator L : L 2 (Ω; R r ) D (Ω; R r ) Lu := d k (A k u) + Cu k=1 is called the symmetric positive operator or the Friedrichs operator, and Lu = f the symmetric positive system or the Friedrichs system.

5 Classical theory Abstract theory Boundary conditions and the trace operator Weak solution of a Friedrichs system belongs only to the graph space W := {u L 2 (Ω; R r ) : Lu L 2 (Ω; R r )}, which is a Hilbert space with L2 (Ω;R r ) + L L L2 (Ω;R r ). Boundary field A ν := d ν k A k L ( Ω; M r (R)), k=1 where ν = (ν 1, ν 2,, ν d ) is the outward unit normal on Ω. Then u A ν u defines the trace operator W H 1 2 ( Ω; R r ). Contribution: K. O. Friedrichs, C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch,...

6 Abstract setting Stationary Friedrichs systems Classical theory Abstract theory A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for the Bijectivity Of Hilbert Operators Related to Friedrichs Systems, Comm. Partial Diff. Eq., 32 (2007), theory written in terms of operators on Hilbert spaces intrinsic criterion for bijectivity of an abstract Friedrichs operator avoiding the question of traces for functions in the graph space Some open questions answered in Nenad Antonić, : Intrinsic boundary conditions for Friedrichs systems, Comm. Partial Diff. Eq. 35 (2010) Nenad Antonić, : Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems, J. Diff. Eq. 250 (2011) Applications: transport equation, stationary diffusion equation, heat equation, Maxwell system in diffusive regime, linearised elasticity,...

7 Assumptions Stationary Friedrichs systems Classical theory Abstract theory L - a real Hilbert space (L L), D L a dense subspace, and L, L : D L linear unbounded operators satisfying (T1) ( ϕ, ψ D) Lϕ ψ L = ϕ Lψ L ; (T2) ( c > 0)( ϕ D) (L + L)ϕ L c ϕ L ; (T3) ( µ 0 > 0)( ϕ D) (L + L)ϕ ϕ L 2µ 0 ϕ 2 L.

8 Classical theory Abstract theory An example: The classical Friedrichs operator Let D := C c (Ω; R r ), L = L 2 (Ω; R r ) and L, L : D L be defined by d Lu := k (A k u) + Cu, k=1 Lu := d k (A k u) + (C + k=1 k=1 d k A k )u, where A k and C are as before (they satisfy (F1) (F2)). Then L and L satisfy (T1) (T3)

9 Extensions Stationary Friedrichs systems Classical theory Abstract theory (D, L ) is an inner product space, with L := L + L L L ; L is the graph norm. Denote by W 0 the completion of D. L, L : D L are continuous with respect to ( L, L )... extension by density to L(W 0, L) Gelfand triple: W 0 L L W 0. Let L L(L, W 0) be the adjoint operator of L : W 0 L: ( u L)( v W 0 ) W 0 L u, v W0 = u Lv L. Therefore L = L W0 Analogously L = L W0 Further extensions... L := L, L := L,... L, L L(L, W 0)... (T)

10 Posing the problem Stationary Friedrichs systems Classical theory Abstract theory Lemma The graph space W := {u L : Lu L} = {u L : Lu L} is a Hilbert space with respect to L. Problem: for given f L find u W such that Lu = f. Find sufficient conditions on V W such that L V isomorphism. : V L is an

11 Boundary operator Stationary Friedrichs systems Classical theory Abstract theory Boundary operator D L(W, W ): W Du, v W := Lu v L u Lv L, u, v W. If L is the classical Friedrichs operator, then for u, v C c (R d ; R r ): W Du, v W = A ν (x)u Γ (x) v Γ (x)ds(x). Let V and Ṽ be subspaces of W that satisfy Γ (V1) ( u V ) W Du, u W 0, ( v Ṽ ) W Dv, v W 0, (V2) V = D(Ṽ )0, Ṽ = D(V ) 0.

12 Well-posedness theorem Classical theory Abstract theory Theorem Under assumptions (T1) (T3) and (V1) (V2), the operators L V : V L and L Ṽ : Ṽ L are isomorphisms. Theorem (Banach Nečas Babuška) Let V and L be two Banach paces, L dual of L and L L(V ; L). Then the following statements are equivalent: a) L is a bijection; b) It holds: ( α > 0)( u V ) Lu L α u V ; ( ) ( v L ) ( u V ) L v, Lu L = 0 = v = 0.

13 Classical theory Abstract theory An example stationary diffusion equation Nenad Antonić,, Marko Vrdoljak: Second-order equations as Friedrichs systems, Nonlinear Analysis: RWA, 14 (2014) We consider the equation div (A u) + cu = f in Ω R d, where f L 2 (Ω), A L (Ω; M d (R)), c L (Ω) satisfy αi A βi, α c β, for some β α > 0. New unknown vector function taking values in R d+1 : [ ] [ ] u σ A u u = u u =. u Then the starting equation can be written as a first-order system { u u + A 1 u σ = 0 div u σ + cu u, = f

14 Classical theory Abstract theory An example stationary diffusion equation (cont.) which is a Friedrichs system with the choice of A k = e k e d+1 + e d+1 e k M d+1 (R), C = [ ] A c The graph space: W = L 2 div (Ω) H1 (Ω). Dirichlet, Neumann and Robin boundary conditions are imposed by the following choice of V and Ṽ : V D = ṼD :=L 2 div(ω) H 1 0(Ω), V N = ṼN :={(u σ, u u ) W : ν u σ = 0}, V R :={(u σ, u u ) W : ν u σ = au u Γ }, Ṽ R :={(u σ, u u ) W : ν u σ = au u Γ }.

15 An example heat equation Classical theory Abstract theory Nenad Antonić,, Marko Vrdoljak: Heat equation as a Friedrichs system, Journal of Mathematical Analysis and Applications 404 (2013), Let Ω R d, T > 0 and Ω T := Ω 0, T. We consider: t u div x (A x u) + cu = f in Ω T. It can be written as Friedrichs system in a similar way as stationary diffusion equation, with the following difference: domain is Ω T R d+1 and A d+1 = e d+1 e d+1 M d+1 (R). { } The graph space: W = u L 2 div (Ω T ) : u u L 2 (0, T ; H 1 (Ω)). Dirichlet boundary condition and zero initial condition: { } V = u W : u u L 2 (0, T ; H 1 0(Ω)), u u (, 0) = 0 a.e. on Ω, { } Ṽ = v W : v u L 2 (0, T ; H 1 0(Ω)), v u (, T ) = 0 a.e. on Ω.

16 Classical theory Abstract theory Homogenisation theory for (classical) Friedrichs systems Developed in, Marko Vrdoljak: Homogenisation theory for Friedrichs systems, Communications on Pure and Applied Analysis, in press. notions of H- and G-convergence compactness result contains homogenisation theory for the stationary diffusion equation and the heat equation

17 Classical theory Abstract theory 1 Stationary Friedrichs systems Classical theory Abstract theory 2 3

18 Non-stationary problem L - real Hilbert space, as before (L L), T > 0 We consider an abstract Cauchy problem in L: (P) { u (t) + Lu(t) = f(t) u(0) = u 0, where f : 0, T L, u 0 L are given, L (not depending on t) satisfies (T1), (T2) and (T3 ) ( ϕ D) (L + L)ϕ ϕ L 0, u : [0, T L is unknown. A priori estimate: t ) ( t [0, T ]) u(t) 2 L e ( u t 0 2 L + f(s) 2 L. 0

19 Semigroup setting Stationary Friedrichs systems Let A : V L L, A := L V Then (P) becomes: (P ) { u (t) Au(t) = f(t) u(0) = u 0. Theorem The operator A is an infinitensimal generator of a C 0 -semigroup on L.

20 Hille-Yosida Stationary Friedrichs systems Theorem A linear operator A : D(A) L L is the infinitesimal generator of a contraction C 0 -semigroup (T (t)) t 0 if and only if where D(A) is dense in X, A is closed, ρ(a) 0, and for every λ > 0 we have R λ (A) 1 λ, ρ(a) := {λ R : λi A : D(A) X is isomorphism}, is a resolvent set of A and is a resolvent of A. R λ (A) := (λi A) 1, λ ρ(a),

21 Existence and uniqueness result Corollary Let L be an operator that satisfies (T1) (T2) and (T3), let V be a subspace of its graph space satisfying (V1) (V2), and f L 1 ( 0, T ; L). Then for every u 0 L the problem (P) has the unique weak solution u C([0, T ]; L) given with u(t) = T (t)u 0 + t 0 T (t s)f(s)ds, t [0, T ], where (T (t)) t 0 is the semigroup generated by A. ( ) If additionally f C([0, T ]; L) W 1,1 ( 0, T ; L) L 1 ( 0, T ; V ) with V equipped with the graph norm and u 0 V, then the above weak solution is the classical solution of (P) on [0, T.

22 Weak solution Stationary Friedrichs systems Theorem Let u 0 L, f L 1 ( 0, T ; L) and let u(t) = T (t)u 0 + be the weak solution of (P). t Then u, Lu, f L 1 ( 0, T ; W 0) and u + Lu = f, in L 1 ( 0, T ; W 0). 0 T (t s)f(s)ds, t [0, T ],

23 Bound on solution Stationary Friedrichs systems From we get: u(t) = T (t)u 0 + t 0 T (t s)f(s)ds, t [0, T ], ( t [0, T ]) u(t) L u 0 L + A priori estimate was: ( t [0, T ]) u(t) 2 L e t ( u 0 2 L + t 0 f(s) L ds. t 0 ) f(s) 2 L.

24 Non-stationary Maxwell system 1/5 Let Ω R 3 be open and bounded with a Lipshitz boundary Γ, µ, ε W 1, (Ω) positive and away from zero, Σ ij L (Ω; M 3 (R)), i, j {1, 2}, and f 1, f 2 L 1 ( 0, T ; L 2 (Ω; R 3 )). We consider a generalized non-stationary Maxwell system (MS) { µ t H + rot E + Σ 11 H + Σ 12 E = f 1 ε t E rot H + Σ 21 H + Σ 22 E = f 2 in 0, T Ω, where E, H : [0, T Ω R 3 are unknown functions. Change of variable [ ] [ ] u1 µh u := =, c := 1 W 1, (Ω), εe µε u 2 turns (MS) to the Friedrichs system t u + Lu = F,

25 Non-stationary Maxwell system 2/5 with A 1 := c, A := c, [ 1 ] µ f 1 A 3 := c, F =, C := ε f (F1) and (F2) are satisfied (with change v := e λt u for large λ > 0, if needed)

26 Non-stationary Maxwell system 3/5 The spaces involved: L = L 2 (Ω; R 3 ) L 2 (Ω; R 3 ), W = L 2 rot(ω; R 3 ) L 2 rot(ω; R 3 ), W 0 = L 2 rot,0(ω; R 3 ) L 2 rot,0(ω; R 3 ) = Cl W C c (Ω; R 6 ), where L 2 rot(ω; R 3 ) is the graph space of the rot operator. The boundary condition ν E Γ = 0 corresponds to the following choice of spaces V, Ṽ W : V = Ṽ = {u W : ν u 2 = 0} = {u W : ν E = 0} = L 2 rot(ω; R 3 ) L 2 rot,0(ω; R 3 ).

27 Non-stationary Maxwell system 4/5 Theorem Let E 0 L 2 rot,0(ω; R 3 ), H 0 L 2 rot(ω; R 3 ) and let f 1, f 2 C([0, T ]; L 2 (Ω; R 3 )) satisfy at least one of the following conditions: f 1, f 2 W 1,1 ( 0, T ; L 2 (Ω; R 3 )); f 1 L 1 ( 0, T ; L 2 rot(ω; R 3 )), f 2 L 1 ( 0, T ; L 2 rot,0(ω; R 3 )). Then the abstract initial-boundary value problem µh + rot E + Σ 11 H + Σ 12 E = f 1 εe rot H + Σ 21 H + Σ 22 E = f 2 E(0) = E 0 H(0) = H 0 ν E Γ = 0,

28 Non-stationary Maxwell system 5/5 Theorem...has unique classical solution given by [ ] [ 1 ] H µ [ I 0 ] µh0 (t) = E 1 T (t) εe0 0 ε I [ 1 ] µ I 0 t + 1 T (t s) 0 ε I 0 [ 1 ] µ f 1 (s) ds, t [0, T ], 1 ε f 2 (s) where (T (t)) t 0 is the contraction C 0 -semigroup generated by L.

29 Other examples Stationary Friedrichs systems Symmetric hyperbolic system d t u + k (A k u) + Cu = f in 0, T R d, k=1 u(0, ) = u 0 Non-stationary div-grad problem t q + p = f 1 in 0, T Ω, Ω R d, 1 c 2 t p + div q = f 2 in 0, T Ω, 0 p Ω = 0, p(0) = p 0, q(0) = q 0 Wave equation { ttu c 2 u = f in 0, T R d u(0, ) = u 0, t u(0, ) = u 1 0.

30 Friedrichs systems in a complex Hilbert space Let L be a complex Hilbert space, L L its antidual, D L, L, L : L L linear operators that satisfy (T1) (T3) (or T3 instead of T3). Technical differences with respect to the real case, but results remain the same... For the classical Friedrichs operator we require (F1) matrix functions A k are selfadjoint: A k = A k, (F2) ( µ 0 > 0) C + C + d k A k 2µ 0 I (ae on Ω), k=1 and again (F1) (F2) imply (T1) (T3).

31 Application to Dirac system 1/2 We consider the Cauchy problem (DS) Lu {}}{ 3 t u + A k k u + Cu = f in 0, T R 3, k=1 u(0) = u 0, where u : [0, T R 3 C 4 is an unknown function, u 0 : R 3 C 4, f : [0, T R 3 C 4 are given, and [ ] [ ] 0 σk c1 I 0 A k :=, k 1..3, C :=, σ k 0 0 c 2 I where σ 1 := [ ] 0 1, σ := are Pauli matrices, and c 1, c 2 L (R 3 ; C). [ ] [ ] 0 i 1 0, σ i 0 3 :=, (F1) (F2)

32 Application to Dirac system 2/2 Theorem Let u 0 W and let f C([0, T ]; L 2 (R 3 ; C 4 )) satisfies at least one of the following conditions: f W 1,1 ( 0, T ; L 2 (R 3 ; C 4 )); f L 1 ( 0, T ; W ). Then the abstract Cauchy problem 3 t u + A k k u + Cu = f k=1 u(0) = u 0 has unique classical solution given with u(t) = T (t)u 0 + t 0 T (t s)f(s)ds, t [0, T ], where (T (t)) t 0 is the contraction C 0 -semigroup generated by L.

33 1 Stationary Friedrichs systems Classical theory Abstract theory 2 3

34 Time-dependent coefficients The operator L depends on t (i.e. the matrix coefficients A k and C depend on t if L is a classical Friedrichs operator): { u (t) + L(t)u(t) = f(t) u(0) = u 0. Semigroup theory can treat time-dependent case, but conditions that ensure existence/uniqueness result are rather complicated to verify...

35 Semilinear problem Consider where f : [0, T L L. { u (t) + Lu(t) = f(t, u(t)) u(0) = u 0, semigroup theory gives existence and uniqueness of solution it requires (locally) Lipshitz continuity of f in variable u if L = L 2 it is not appropriate assumption, as power functions do not satisfy it; L = L is better...

36 Banach space setting Let L be a reflexive complex Banach space, L its antidual, D L, L, L : D L linear operators that satisfy a modified versions of (T1) (T3), e.g. (T1) ( ϕ, ψ D) L Lϕ, ψ L = L Lψ, φ L. Technical differences with Hilbert space case, but results remain essentially the same for the stationary case... Problems: in the classical case (F1) (F2) need not to imply (T2) and (T3): instead of (T3) we get L p (L + L)ϕ, ϕ L p ϕ L 2 for semigroup treatment of non-stationary case we need to have L : D L L

37 Other... Stationary Friedrichs systems regularity of the solution application to new examples numerical treatment