Multi-d shock waves and surface waves

Transcription

1 Multi-d shock waves and surface waves S. Benzoni-Gavage University of Lyon (Université Claude Bernard Lyon 1 / Institut Camille Jordan) HYP2008 conference, June 11, / 26

2 Outline Theory Eamples 1 Theory Eamples / 26

3 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. Φ= / 26

4 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. Φ= 0 1 Basic assumption: hyperbolicity in t-direction, i.e. for all u U R n, the matri A 0 (u) := df 0 (u) is nonsingular, and for all ν R d, the matri A 0 (u) 1 A j (u) ν j only has real semisimple eigenvalues. 3 / 26

5 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. Φ= 0 1 A 0 (u, ν) := A 0 (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) 3 / 26

6 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. Φ= 0 1 A 0 (u, ν) := A 0 (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) Shock is noncharacteristic iff both matrices A 0 (u ±, Φ) are nonsingular along Φ = 0. 3 / 26

7 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. Φ= 0 1 A 0 (u, ν) := A 0 (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) Shock is classical (or Laian) iff dime u (A 0 (u, Φ)) + dime s (A 0 (u +, Φ)) = n + 1, dime u (A 0 (u +, Φ) + dime s (A 0 (u, Φ)) = n 1. 3 / 26

8 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. Φ= 0 1 A 0 (u, ν) := A 0 (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) Shock is undercompressive iff dime u (A 0 (u, Φ)) + dime s (A 0 (u +, Φ)) = n + 1 p, dime u (A 0 (u +, Φ) + dime s (A 0 (u, Φ)) = n 1 + p. 3 / 26

9 Theory Eamples General equations for a shock wave t t f 0 (u) + j f j (u) = 0 n, Φ(t, ) 0, Φ= 0 d [f 0 (u)] t Φ + [f j (u)] j Φ = 0 n, Φ(t, ) = 0. [g 0 (u)] t Φ + [g j (u)] j Φ = 0 p, Φ(t, ) = 0. 1 A 0 (u, ν) := A 0 (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) Shock is undercompressive iff dime u (A 0 (u, Φ)) + dime s (A 0 (u +, Φ)) = n + 1 p, dime u (A 0 (u +, Φ) + dime s (A 0 (u, Φ)) = n 1 + p. 3 / 26

10 Linear analysis Theory Eamples Ref. planar shock t = σ t d d 1 [Lopatinskĭı 70], [Kreiss 70], [Blokhin 82], [Majda 83], [Freistühler 98]. 4 / 26

11 Linear analysis Theory Eamples Ref. planar shock t Change frame = σ = 0. d 1 [Lopatinskĭı 70], [Kreiss 70], [Blokhin 82], [Majda 83], [Freistühler 98]. 4 / 26

12 Linear analysis Theory Eamples Ref. planar shock t d Change frame = σ = 0. Change coordinates (t, ) (t, y) := (t, 1,..., d 1, Φ(t, )), Φ(t, ) = d χ(t, 1,..., d 1 ). 1 [Lopatinskĭı 70], [Kreiss 70], [Blokhin 82], [Majda 83], [Freistühler 98]. 4 / 26

13 Linear analysis Theory Eamples Ref. planar shock t d Change frame = σ = 0. Change coordinates (t, ) (t, y) := (t, 1,..., d 1, Φ(t, )), Φ(t, ) = d χ(t, 1,..., d 1 ). Linearize eqns about (u, χ) = (u, 0), u := u ±, y d 0. 1 [Lopatinskĭı 70], [Kreiss 70], [Blokhin 82], [Majda 83], [Freistühler 98]. 4 / 26

14 Normal modes analysis Theory Eamples Transmission problem: { A0 (u) t u + A j (u) j u = 0 n, y d 0, [F 0 (u)] t χ + [F j (u)] j χ = [df d (u) u], y d = 0. 5 / 26

15 Normal modes analysis Theory Eamples Transmission problem: { A0 (u) t u + A j (u) j u = 0 n, y d 0, [F 0 (u)] t χ + [F j (u)] j χ = [df d (u) u], y d = 0. Fourier-Laplace transform (t, ˇy) (τ, ˇη) shooting ODE problem. 5 / 26

16 Normal modes analysis Theory Eamples Transmission problem: { A0 (u) t u + A j (u) j u = 0 n, y d 0, [F 0 (u)] t χ + [F j (u)] j χ = [df d (u) u], y d = 0. Fourier-Laplace transform (t, ˇy) (τ, ˇη) shooting ODE problem. A d (u, ν) := A d (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) 5 / 26

17 Normal modes analysis Theory Eamples Transmission problem: { A0 (u) t u + A j (u) j u = 0 n, y d 0, [F 0 (u)] t χ + [F j (u)] j χ = [df d (u) u], y d = 0. Fourier-Laplace transform (t, ˇy) (τ, ˇη) shooting ODE problem. A d (u, ν) := A d (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) Normal modes: χ = X e τt+iη j y j, u = U(y d ) e τt+iη j y j with U L 2 (R), Re (τ) > 0, U(0+) E u (A d (u, τ, i ˇη)) and U(0 ) E s (A d (u, τ, i ˇη)). 5 / 26

18 Normal modes analysis Theory Eamples Transmission problem: { A0 (u) t u + A j (u) j u = 0 n, y d 0, [F 0 (u)] t χ + [F j (u)] j χ = [df d (u) u], y d = 0. Fourier-Laplace transform (t, ˇy) (τ, ˇη) shooting ODE problem. A d (u, ν) := A d (u) 1 (A 0 (u)ν 0 + A j (u) ν j ) Neutral modes of finite energy, or surface waves: χ = X e iη 0t+iη j y j, u = U(y d ) e iη 0t+iη j y j with still U L 2 (R), U(0+) E u (A d (u, iη 0, i ˇη)) and U(0 ) E s (A d (u, iη 0, i ˇη)). 5 / 26

19 Surface waves Theory Eamples t d 1 u d 1 6 / 26

20 Surface waves Theory Eamples t Isotropic elasticity tt u = λ u + (λ + µ) divu, 2 > 0, 2 u u 2 = 0, 2 = 0, d µ 1 u 1 + (2λ + µ) 2 u 2 = 0, 2 = 0. 1 u d [Rayleigh1885] (see also [Serre 06]) 1 6 / 26

21 Surface waves Theory Eamples t Isotropic elasticity tt u = λ u + (λ + µ) divu, 2 > 0, 2 u u 2 = 0, 2 = 0, d µ 1 u 1 + (2λ + µ) 2 u 2 = 0, 2 = 0. 1 u d For λ > 0, λ + µ > 0, Rayleigh waves, or Surface Acoustic Waves, of speed less than λ. [Rayleigh1885] (see also [Serre 06]) 1 6 / 26

22 Surface waves Theory Eamples t Classical shocks in gas dynamics [Bethe 42], [D yakov 54], [Iordanskĭı 57], [Kontorovič 58], [Erpenbeck 62], [Majda 83], [Blokhin 82]. d 1 u d 1 [Menikoff Plohr 89], [SBG Serre 07]. [Jenssen-Lyng 04], 6 / 26

23 Surface waves Theory Eamples t Classical shocks in gas dynamics [Bethe 42], [D yakov 54], [Iordanskĭı 57], [Kontorovič 58], [Erpenbeck 62], [Majda 83], [Blokhin 82]. 1 u d d There eist neutral modes iff 1 M < k 1 + M 2 (r 1), where M = Mach number behind the shock, r = v p /v b with v p,b = volume past/behind the shock, k = 2 + M 2 (v b v p) T p s. 1 [Menikoff Plohr 89], [SBG Serre 07]. [Jenssen-Lyng 04], 6 / 26

24 Surface waves Theory Eamples t Phase boundaries [SBG 98-99], [SBG Freistühler 04] d 1 u d 1 6 / 26

25 Surface waves Theory Eamples t Phase boundaries [SBG 98-99], [SBG Freistühler 04] 1 u d For nondissipative subsonic phase boundaries there eist surface waves, of speed less than u b u p. d 1 6 / 26

26 Outline 1 Theory Eamples / 26

27 Constant coefficients linear problem Interior operator L(u) := A 0 (u) t + A j (u) j Boundary operator B(u) := [F 0 (u)] t + [F j (u)] j [df d (u) ] 8 / 26

28 Constant coefficients linear problem Interior operator L(u) := A 0 (u) t + A j (u) j Boundary operator B(u) := [F 0 (u)] t + [F j (u)] j [df d (u) ] Maimal a priori estimate γ e γt u 2 L 2 + e γt u yd =0 2 L 2 + e γt χ 2 H 1 γ 1 γ e γt L(u)u 2 L 2 + e γt B(u)(χ, u) 2 L 2 8 / 26

29 Constant coefficients linear problem Interior operator L(u) := A 0 (u) t + A j (u) j Boundary operator B(u) := [F 0 (u)] t + [F j (u)] j [df d (u) ] Maimal a priori estimate γ e γt u 2 L 2 + e γt u yd =0 2 L 2 + e γt χ 2 H 1 γ 1 γ e γt L(u)u 2 L 2 + e γt B(u)(χ, u) 2 L 2 OK under uniform Kreiss-Lopatinskĭı condition, i.e. without neutral modes. (Proof based on Kreiss symmetrizers technique.) 8 / 26

30 Constant coefficients linear problem Interior operator L(u) := A 0 (u) t + A j (u) j Boundary operator B(u) := [F 0 (u)] t + [F j (u)] j [df d (u) ] A priori estimate with loss of derivatives γ e γt u 2 L 2 + e γt u yd =0 2 L 2 + e γt χ 2 H 1 γ 1 γ 3 e γt L(u)u 2 L 2 (R + ;H 1 γ ) + 1 γ 2 e γt B(u)(χ, u) 2 H 1 γ 9 / 26

31 Constant coefficients linear problem Interior operator L(u) := A 0 (u) t + A j (u) j Boundary operator B(u) := [F 0 (u)] t + [F j (u)] j [df d (u) ] A priori estimate with loss of derivatives γ e γt u 2 L 2 + e γt u yd =0 2 L 2 + e γt χ 2 H 1 γ 1 γ 3 e γt L(u)u 2 L 2 (R + ;H 1 γ ) + 1 γ 2 e γt B(u)(χ, u) 2 H 1 γ Takes into account neutral modes. (Proof still based on Kreiss symmetrizers technique [Coulombel 02], [Sablé-Tougeron 88].) 9 / 26

32 Fully nonlinear problem Local-in-time eistence of smooth solutions 10 / 26

33 Fully nonlinear problem Local-in-time eistence of smooth solutions under uniform Kreiss Lopatinskĭı condition [Majda 83], [Blokhin 82], [Métivier et al ], 10 / 26

34 Fully nonlinear problem Local-in-time eistence of smooth solutions under uniform Kreiss Lopatinskĭı condition [Majda 83], [Blokhin 82], [Métivier et al ], under mere Kreiss Lopatinskĭı condition [Coulombel Secchi 08]: with neutral modes and characteristic modes ; application to subsonic phase boundaries and compressible 2d-vorte sheets. (Proof using Nash Moser iteration scheme.) 10 / 26

35 Outline 1 Theory Eamples / 26

36 Fully nonlinear problem { A0 (u) t u + A j (u) j u + A d (u, χ) d u = 0 n, y d 0, [F 0 (u)] t χ + [F j (u)] j χ = [F d (u)], y d = 0. A d (u, χ) := A d (u) A 0 (u) t χ A j (u) j χ 12 / 26

37 Fully nonlinear problem { A0 (u) t u + A j (u) j u + A d (u, χ) d u = 0 n, y d 0, J χ + h(u) = 0 n+p, y d = 0. A d (u, χ) := A d (u) A 0 (u) t χ A j (u) j χ 1... J = / 26

38 Fully nonlinear problem { A0 (u) t u + A j (u) j u + A d (u, χ) d u = 0 n, y d 0, J χ + h(u) = 0 n+p, y d = 0. Asymptotic epansion u = u + ε u(η 0 t + η j y j, y d, εt) + ε 2 ü(η 0 t + η j y j, y d, εt) + h.o.t. χ = ε χ(η 0 t + η j y j, y d, εt) + ε 2 χ(η 0 t + η j y j, εt) + h.o.t. [SBG-Rosini 08], [Hunter 89], [Parker 88]. 12 / 26

39 Approimate problems ξ := η 0 t + η j y j, z := y d, τ := εt. First order { (η0 A 0 (u) + η j A j (u)) ξ u + A d (u) z u = 0 n, z 0, Jη ξ χ + dh(u) u = 0 n+p, z = 0, 13 / 26

40 Approimate problems ξ := η 0 t + η j y j, z := y d, τ := εt. First order { (η0 A 0 (u) + η j A j (u)) ξ u + A d (u) z u = 0 n, z 0, Jη ξ χ + dh(u) u = 0 n+p, z = 0, Second order { (η0 A 0 (u) + η j A j (u)) ξ ü + A d (u) z ü = Ṁ, z 0, Jη ξ χ + dh(u) ü = Ġ, z = 0, 13 / 26

41 Approimate problems ξ := η 0 t + η j y j, z := y d, τ := εt. First order { (η0 A 0 (u) + η j A j (u)) ξ u + A d (u) z u = 0 n, z 0, Jη ξ χ + dh(u) u = 0 n+p, z = 0, Second order { (η0 A 0 (u) + η j A j (u)) ξ ü + A d (u) z ü = Ṁ, z 0, Jη ξ χ + dh(u) ü = Ġ, z = 0, Ṁ := A 0 (u) τ u + (η 0 da 0 (u) + η j da j (u)) u ξ u + da d (u) u z u ( ξ χ)(η 0 A 0 (u) + η j A j (u)) z u 13 / 26

42 Approimate problems ξ := η 0 t + η j y j, z := y d, τ := εt. First order { (η0 A 0 (u) + η j A j (u)) ξ u + A d (u) z u = 0 n, z 0, Jη ξ χ + dh(u) u = 0 n+p, z = 0, Second order { (η0 A 0 (u) + η j A j (u)) ξ ü + A d (u) z ü = Ṁ, z 0, Jη ξ χ + dh(u) ü = Ġ, z = 0, Ġ := ( τ χ)e d2 h(u) ( u, u). 13 / 26

43 Transformation of approimate problems Fourier transform ξ k, First order { ik (η0 A 0 (u) + η j A j (u)) u + A d (u) z u = 0 n, z 0, ik Jη χ + dh(u) u = 0 n+p, z = 0, Second order { ik (η0 A 0 (u) + η j A j (u)) ü + A d (u) z ü = Ṁ, z 0, ik Jη χ + dh(u) ü = Ġ, z = / 26

44 Transformation of approimate problems Fourier transform ξ k, elimination of χ, χ. First order { ik (η0 A 0 (u) + η j A j (u)) u + A d (u) z u = 0 n, z 0, C(u; η) u = 0 n+p 1, z = 0, Second order { ik (η0 A 0 (u) + η j A j (u)) ü + A d (u) z ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = / 26

45 Transformation of approimate problems Fourier transform ξ k, elimination of χ, χ. First order { L(u; kη) u = 0n, z 0, C(u; η) u = 0 n+p 1, z = 0, Second order { L(u; kη) ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = 0. L(u; η) := iη 0 A 0 (u) + iη j A j (u) + A d (u) z. 14 / 26

46 Resolution of approimate problems First order level Eistence of linear surface wave = L 2 (dz) solution u 1 of { L(u; η) u1 = 0 n, z 0, C(u; η) u 1 = 0 n+p 1 z = / 26

47 Resolution of approimate problems First order level Eistence of linear surface wave = L 2 (dz) solution u 1 of { L(u; η) u1 = 0 n, z 0, C(u; η) u 1 = 0 n+p 1 z = 0. Homogeneity = other square integrable solutions of first order system of the form u(k, z, τ) = W (k, τ) u 1 (kz), k > / 26

48 Resolution of approimate problems First order level Eistence of linear surface wave = L 2 (dz) solution u 1 of { L(u; η) u1 = 0 n, z 0, C(u; η) u 1 = 0 n+p 1 z = 0. Homogeneity = other square integrable solutions of first order system of the form u(k, z, τ) = W (k, τ) u 1 (kz), k > 0. Amplitude function: F 1 (W ) =: w(ξ, τ). 15 / 26

49 Resolution of approimate problems First order level Eistence of linear surface wave = L 2 (dz) solution u 1 of { L(u; η) u1 = 0 n, z 0, C(u; η) u 1 = 0 n+p 1 z = 0. Homogeneity = other square integrable solutions of first order system of the form u(k, z, τ) = W (k, τ) u 1 (kz), k > 0. Amplitude function: F 1 (W ) =: w(ξ, τ). Second order level Solvability condition by means of an adjoint problem. 15 / 26

50 Solvability of second order problem ( ) { L(u; kη) ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = 0, 16 / 26

51 Solvability of second order problem ( ) { L(u; kη) ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = 0, L(u; η) := iη 0 A 0 (u) + iη j A j (u) + A d (u) z, 16 / 26

52 Solvability of second order problem ( ) { L(u; kη) ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = 0, L(u; η) := iη 0 A 0 (u) + iη j A j (u) + A d (u) z, C(u; η)u = C + u(0+) C u(0 ), 16 / 26

53 Solvability of second order problem ( ) { L(u; kη) ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = 0, L(u; η) := iη 0 A 0 (u) + iη j A j (u) + A d (u) z, C(u; η)u = C + u(0+) C u(0 ), ( ) ( Ad (u ) 0 D = 0 A d (u + ) D+ ) N + P ( C C + ). 16 / 26

54 Solvability of second order problem ( ) { L(u; kη) ü = Ṁ, z 0, C(u; η)ü = T (η)ġ, z = 0, L(u; η) := iη 0 A 0 (u) + iη j A j (u) + A d (u) z, C(u; η)u = C + u(0+) C u(0 ), ( ) ( ) Ad (u ) 0 D = 0 A d (u + ) D+ N + P ( C C + ). There eists a L 2 (dz) solution ü of ( ) iff v Ṁ dz + (v(0 ) v(0+) )PT Ġ = 0, with v solution of { L(u; kη) v = 0 n, z 0, D + v(0+) D v(0 ) = 0 n p+1, z = / 26

55 Resulting amplitude equation Nonlocal generalisation of Burgers equation: τ w + ξ Q[w] = 0, F (Q[w])(k) = + Λ(k l, l)ŵ(k l)ŵ(l)dl. with piecewise smooth kernel Λ, homogeneous degree / 26

56 Resulting amplitude equation Nonlocal generalisation of Burgers equation: F (Q[w])(k) = τ w + ξ Q[w] = 0, + Λ(k l, l)ŵ(k l)ŵ(l)dl. with piecewise smooth kernel Λ, homogeneous degree 0. Recover classical inviscid Burgers equation if Λ 1/2 (arises in case of neutral modes of infinite energy [Artola-Majda 87]). 17 / 26

57 Nonlocal Burgers equations In Fourier variables: τ ŵ + ik + Λ(k l, l)ŵ(k l, τ)ŵ(l, τ)dl = 0. Eistence of smooth solutions? Well-posedness? 18 / 26

58 Nonlocal Burgers equations In Fourier variables: τ ŵ + ik + Λ(k l, l)ŵ(k l, τ)ŵ(l, τ)dl = 0. Eistence of smooth solutions? Well-posedness? Properties of Λ: 18 / 26

59 Nonlocal Burgers equations In Fourier variables: τ ŵ + ik + Λ(k l, l)ŵ(k l, τ)ŵ(l, τ)dl = 0. Eistence of smooth solutions? Well-posedness? Properties of Λ: Λ(k, l) = Λ(l, k) 18 / 26

60 Nonlocal Burgers equations In Fourier variables: τ ŵ + ik + Λ(k l, l)ŵ(k l, τ)ŵ(l, τ)dl = 0. Eistence of smooth solutions? Well-posedness? Properties of Λ: Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) 18 / 26

61 Nonlocal Burgers equations In Fourier variables: τ ŵ + ik + Λ(k l, l)ŵ(k l, τ)ŵ(l, τ)dl = 0. Eistence of smooth solutions? Well-posedness? Properties of Λ: Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) Λ(1, 0 ) = Λ(1, 0+) 18 / 26

62 Nonlocal Burgers equations In Fourier variables: τ ŵ + ik + Λ(k l, l)ŵ(k l, τ)ŵ(l, τ)dl = 0. Eistence of smooth solutions? Well-posedness? Properties of Λ: Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) Λ(k + ξ, ξ) = Λ(k, ξ) 18 / 26

63 Hamiltonian nonlocal Burgers equations Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) Λ(k + ξ, ξ) = Λ(k, ξ) = Hamiltonian structure : H[w] := 1 3 τ w + δh[w] = 0, Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dk dl. 19 / 26

64 Hamiltonian nonlocal Burgers equations Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) Λ(k + ξ, ξ) = Λ(k, ξ) = Hamiltonian structure : H[w] := 1 3 τ w + δh[w] = 0, Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dk dl. = Local eistence of smooth periodic solutions [Hunter 06] (also see [Alì Hunter Parker 02]). 19 / 26

65 Stable nonlocal Burgers equations Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) Λ(1, 0 ) = Λ(1, 0+) = a priori estimates, 20 / 26

66 Stable nonlocal Burgers equations Λ(k, l) = Λ(l, k) Λ( k, l) = Λ(k, l) Λ(1, 0 ) = Λ(1, 0+) = a priori estimates, and eventually local H 2 well-posedness [SBG 08]. 20 / 26

67 A priori estimates Local Burgers: d dτ ( n ξ w)2 ξ w L ( n ξ w)2. 21 / 26

68 A priori estimates Local Burgers: d dτ ( n ξ w)2 ξ w L ( n ξ w)2. Nonlocal Burgers: d ( ξ n dτ w)2 F ( ξ w) L 1 ( n ξ w)2. 21 / 26

69 A priori estimates L 2 estimate (n = 0) : d w 2 dξ = d ŵ 2 dk = dτ dτ ( ) 2 Re i k Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk 22 / 26

70 A priori estimates L 2 estimate (n = 0) : d w 2 dξ = d ŵ 2 dk = dτ dτ ( ) 2 Re i k Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk 2 Λ L ŵ 2 L 2 by Fubini and Cauchy-Schwarz! kŵ(k) dk 22 / 26

71 A priori estimates H 1 estimate (n = 1) : d ( ξ w) 2 dξ = d k 2 ŵ 2 dk = dτ dτ ( ) 2 Re i k 3 Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk 23 / 26

72 A priori estimates H 1 estimate (n = 1) : d ( ξ w) 2 dξ = d k 2 ŵ 2 dk = dτ dτ ( ) 2 Re i k 3 Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk = ( 4 Re ) i k 2 (k l) Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk 23 / 26

73 A priori estimates H 1 estimate (n = 1) : d ( ξ w) 2 dξ = d k 2 ŵ 2 dk = dτ dτ ( ) 2 Re i k 3 Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk = ( 4 Re ) i k 2 (k l) Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk. k l... Λ L ξ w L 1 ξ w 2 L / 26

74 A priori estimates H 1 estimate (cont.) ( ) Re i k 2 (k l) Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk = k > l i i k > l k > l k 2 (k l) Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk k 2 (k l) Λ(l k, l)ŵ(l k)ŵ( l)ŵ(k) dldk. 24 / 26

75 A priori estimates H 1 estimate (cont.) ( ) Re i k 2 (k l) Λ(k l, l)ŵ(k l)ŵ(l)ŵ( k) dldk k > l i k > l k(k l) ((k l) Λ(l k, l) kλ(k, l)) ŵ(l k)ŵ( l)ŵ(k) dldk. after change of variables (k, l) (k l, l) in first integral. = 24 / 26

76 Local well-posedness Theorem ([SBG 08]) If Λ is smooth outside the lines k = 0, l = 0, and k + l = 0, homogeneous degree zero, preserves real-valued functions, and satisfies the stability condition Λ(1, 0 ) = Λ( 1, 0 ), then for all w 0 H 2 (R) there eists T > 0 and a unique solution w C (0, T ; H 2 (R)) C 1 (0, T ; H 1 (R)) such that w(0) = w 0 of the nonlocal Burgers equation of kernel Λ, and the mapping is continuous. H 2 (R) C (0, T ; H 2 (R)) w 0 w 25 / 26

77 Blow-up criterion The solution w can be etended beyond T provided that T 0 F ( ξw) L 1 is finite. 26 / 26

78 Blow-up criterion The solution w can be etended beyond T provided that T 0 F ( ξw) L 1 is finite. Applications 26 / 26

79 Blow-up criterion The solution w can be etended beyond T provided that T 0 F ( ξw) L 1 is finite. Applications elasticity: Λ(k + ξ, ξ) = Λ(k, ξ), 26 / 26

80 Blow-up criterion The solution w can be etended beyond T provided that T 0 F ( ξw) L 1 is finite. Applications elasticity: Λ(k + ξ, ξ) = Λ(k, ξ), phase boundaries: Λ(1, 0 ) Λ(1, 0+)! 26 / 26