Regularity of weakly well posed characteristic boundary value - PowerPoint PPT Presentation

Introduction From IBVPs to BVPs The main result Regularity of weakly well posed characteristic boundary value problems Alessandro Morando Department of Mathematics University of Brescia Joint work with P. Secchi Intensive Research Month on Hyperbolic Conservation Laws and Fluid Dynamics Department of Mathematics, University of Parma February 1-28, 2010 A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction From IBVPs to BVPs The main result Plan 1 Introduction Characteristic IBVP for hyperbolic systems Characteristic free boundary problems 2 From IBVPs to BVPs Problem of regularity The general strategy Reduction to a BVP 3 The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Plan 1 Introduction Characteristic IBVP for hyperbolic systems Characteristic free boundary problems 2 From IBVPs to BVPs Problem of regularity The general strategy Reduction to a BVP 3 The main result Tangential Regularity of the BVP Tangential operators Tangential Commutator Normal Commutator A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Characteristic IBVP for hyperbolic systems Consider the problem   Lu = F in Q T ,  Mu = G on Σ T ,   u | t =0 = f in Ω , where + := { x = ( x 1 , x 2 , . . . , x n ) ∈ R n : x 1 > 0 } , Ω := R n Q T := Ω × (0 , T ) , Σ T := ∂ Ω × (0 , T ) L := ∂ t + � n j =1 A j ( x, t ) ∂ x j + B ( x, t ) , A j , B ∈ M N × N M = M ( x, t ) ∈ M d × N , rank ( M ) = d (maximal rank) u ( x, t ) ∈ R N , F ( x, t ) ∈ R N , f ( x ) ∈ R N , G ( x, t ) ∈ R d A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Characteristic IBVP for hyperbolic systems Consider the problem   Lu = F in Q T ,  Mu = G on Σ T ,   u | t =0 = f in Ω , where + := { x = ( x 1 , x 2 , . . . , x n ) ∈ R n : x 1 > 0 } , Ω := R n Q T := Ω × (0 , T ) , Σ T := ∂ Ω × (0 , T ) L := ∂ t + � n j =1 A j ( x, t ) ∂ x j + B ( x, t ) , A j , B ∈ M N × N M = M ( x, t ) ∈ M d × N , rank ( M ) = d (maximal rank) u ( x, t ) ∈ R N , F ( x, t ) ∈ R N , f ( x ) ∈ R N , G ( x, t ) ∈ R d A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Characteristic boundary The boundary ∂ Ω is characteristic with constant multiplicity if the boundary matrix n � A ν := A j ν j j =1 is singular with constant rank 1 ≤ r = rank A ν < N at ∂ Ω ( ν = ν ( x ) outward normal vector to ∂ Ω ). When Ω = { x 1 > 0 } then A ν = − A 1 | ∂ Ω (since ν = ( − 1 , 0 , . . . , 0) ) d = # { positive eigenvalues of A 1 at ∂ Ω } Full regularity (existence in usual Sobolev spaces H m (Ω) ) can’t be expected, in general, because of the possible loss of normal regularity at ∂ Ω . [Tsuji, Proc. Japan Acad. 1972], MHD [Ohno & Shirota, ARMA 1998]. A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Characteristic boundary The boundary ∂ Ω is characteristic with constant multiplicity if the boundary matrix n � A ν := A j ν j j =1 is singular with constant rank 1 ≤ r = rank A ν < N at ∂ Ω ( ν = ν ( x ) outward normal vector to ∂ Ω ). When Ω = { x 1 > 0 } then A ν = − A 1 | ∂ Ω (since ν = ( − 1 , 0 , . . . , 0) ) d = # { positive eigenvalues of A 1 at ∂ Ω } Full regularity (existence in usual Sobolev spaces H m (Ω) ) can’t be expected, in general, because of the possible loss of normal regularity at ∂ Ω . [Tsuji, Proc. Japan Acad. 1972], MHD [Ohno & Shirota, ARMA 1998]. A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Characteristic boundary The boundary ∂ Ω is characteristic with constant multiplicity if the boundary matrix n � A ν := A j ν j j =1 is singular with constant rank 1 ≤ r = rank A ν < N at ∂ Ω ( ν = ν ( x ) outward normal vector to ∂ Ω ). When Ω = { x 1 > 0 } then A ν = − A 1 | ∂ Ω (since ν = ( − 1 , 0 , . . . , 0) ) d = # { positive eigenvalues of A 1 at ∂ Ω } Full regularity (existence in usual Sobolev spaces H m (Ω) ) can’t be expected, in general, because of the possible loss of normal regularity at ∂ Ω . [Tsuji, Proc. Japan Acad. 1972], MHD [Ohno & Shirota, ARMA 1998]. A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Generally speaking, one normal derivative (w.r.t. ∂ Ω ) is controlled by two tangential derivatives. Natural function space is the weighted anisotropic Sobolev space H m ∗ (Ω) := { u ∈ L 2 (Ω) : Z α ∂ k x 1 u ∈ L 2 (Ω) , | α | + 2 k ≤ m } , where Z α := Z α 1 1 . . . Z α n n , α = ( α 1 , . . . , α n ) , Z 1 = x 1 ∂ x 1 and Z j = ∂ x j for j = 2 , . . . , n, if Ω = { x 1 > 0 } . [Chen Shuxing, Chinese Ann. Math. 1982], [Yanagisawa & Matsumura, CMP 1991]. back to H m tan back to m = 1 A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Kreiss-Lopatinskii condition Consider the BVP � in { x 1 > 0 } , Lu = F , Mu = G , on { x 1 = 0 } . L := ∂ t + � n j =1 A j ∂ x j , hyperbolic operator (with eigenvalues of constant multiplicity); A j ∈ M N × N , j = 1 , . . . , n , and det A 1 � = 0 (i.e. non characteristic boundary); M ∈ M d × N , rank ( M ) = d = # { positive eigenvalues of A 1 } . A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result Kreiss-Lopatinskii condition Consider the BVP � in { x 1 > 0 } , Lu = F , Mu = G , on { x 1 = 0 } . L := ∂ t + � n j =1 A j ∂ x j , hyperbolic operator (with eigenvalues of constant multiplicity); A j ∈ M N × N , j = 1 , . . . , n , and det A 1 � = 0 (i.e. non characteristic boundary); M ∈ M d × N , rank ( M ) = d = # { positive eigenvalues of A 1 } . A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result • Let u = u ( x 1 , x ′ , t ) ( x ′ = ( x 2 , . . . , x n ) ) be a solution to ( ?? ) for F = 0 and G = 0 . u ( x 1 , η, τ ) be Fourier-Laplace transform of u w.r.t. x ′ • Let � u = � and t respectively ( η and τ dual variables of x ′ and t respectively). • � u solves the ODE problem � d b u dx 1 = A ( η, τ ) � u , x 1 > 0 , M � u (0) = 0 , � � n � where A ( η, τ ) := − ( A 1 ) − 1 τI n + i A j η j . j =2 A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result • Let u = u ( x 1 , x ′ , t ) ( x ′ = ( x 2 , . . . , x n ) ) be a solution to ( ?? ) for F = 0 and G = 0 . u ( x 1 , η, τ ) be Fourier-Laplace transform of u w.r.t. x ′ • Let � u = � and t respectively ( η and τ dual variables of x ′ and t respectively). • � u solves the ODE problem � d b u dx 1 = A ( η, τ ) � u , x 1 > 0 , M � u (0) = 0 , � � n � where A ( η, τ ) := − ( A 1 ) − 1 τI n + i A j η j . j =2 A. Morando (Brescia University) Characteristic Hyperbolic BVP’s

Introduction Characteristic IBVP for hyperbolic systems From IBVPs to BVPs Characteristic free boundary problems The main result • Let u = u ( x 1 , x ′ , t ) ( x ′ = ( x 2 , . . . , x n ) ) be a solution to ( ?? ) for F = 0 and G = 0 . u ( x 1 , η, τ ) be Fourier-Laplace transform of u w.r.t. x ′ • Let � u = � and t respectively ( η and τ dual variables of x ′ and t respectively). • � u solves the ODE problem � d b u dx 1 = A ( η, τ ) � u , x 1 > 0 , M � u (0) = 0 , � � n � where A ( η, τ ) := − ( A 1 ) − 1 τI n + i A j η j . j =2 A. Morando (Brescia University) Characteristic Hyperbolic BVP’s