Free Boundary Problems for Nonlinear Wave Equations: Interacting Shocks

Transcription

1 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 1 Free Boundary Problems for Nonlinear Wave Equations: Interacting Shocks Barbara Lee Keyfitz a Department of Mathematics University of Houston Houston, Texas joint work with Sunčica Čanić and Eun Heui Kim computations by Alexander Kurganov a Research supported by the Department of Energy, grant DE-FG ER25222, and by the Texas Advanced Research Program.

2 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 2 Similarity Analysis of Two-Dimensional Systems U t +F (U) x +G(U) y =, ( ) Data: U(x, y, ) = f Similarity Variables: x y U R n y ξ = x t, η = y t U = U(ξ, η) x Reduced System in Two Variables ξ (F ξu) + η (G ηu) F ξ + G η = 2U Sectorially Constant Data Method: resolve 1-D far-field discontinuities; solve as IV/BVP in 2-D Type Changes: hyperbolic in far field; subsonic region near origin

3 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 3 Acoustic-type Structure: Degenerate/Non-degenerate Characteristics U t + AU x + BU y = ; det Iτ + Aλ + Bµ = (n 2 l i σ ) σ T Q N σ i=1 6 4 Characteristic Normals σ=(ξ,η,τ) Acoustic Envelopes in Physical Space 3 Shear 2 Acoustic σ = (τ, λ, µ) ( (A ξi) ξ + (B ηi) η ) U = τ ν 2 2 µ Shear 2 t y x n 2 i=1 CHANGE OF TYPE THEOREM Reduced equation hyperbolic iff x = (1, ξ, η) outside acoustic wave cone C W = {x T Q 1 N x = }. RP in dim CP w. data at Ξ = (ξ, η) dual vector α = (α, β) l i ( α Ξ, α, β) q(σ( α, Ξ), U) }{{} q( α,ξ,u) η ξ SUBSONIC REGION: ELLIPTIC OR MIXED DEGENERATE CHARACTERISTICS τ τ + NONDEGENERATE SPACELIKE CURVE Ξ CHARACTERISTICS SUPERSONIC REGION: HYPERBOLIC L

4 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 4 Far Field Solution and Wave Interactions Quasi-One-Dimensional Riemann Problems in Hyperbolic Region Shock-shock 6 Rarefaction-rarefaction 4 Rarefaction-shock 2 No Q-1-D solns for some probs Mach stems Sonic lines c c 1 c 2 c a c b c c Sonic bdry not det. a priori Current effort: examine simplified Nonlinear Wave System Nonlinear and Linear parts decouple & Linear waves stationary

5 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 5 Comparison: Isentropic Gas Dynamics & NLWS Isentropic Gas Dynamics: p = ρ γ /γ ρ t + (ρu) x + (ρv) y = (ρu) t + (ρu 2 + p) x + (ρuv) y = (ρv) t + (ρuv) x + (ρv 2 + p) y = Nonlinear Wave System: ρ t + m x + n y = m t + p x = n t + p y = ( Second-order equation ) for nonlinear char. variable (ρ): (c 2 (ρ) U 2 ( ) )ρ ξ UV ρ η (c 2 (ρ) ξ 2 )ρ ξ ξηρ η + ( (c 2 (ρ) V 2 )ρ η UV ρ ξ )η +(V u η Uv η )ρ ξ + (Uv ξ V u ξ )ρ η +2(v ξ u η u ξ v η )ρ = U = u ξ, V = v η ( pseudo-vel. ) ξ + ( (c 2 (ρ) η 2 )ρ η ξηρ ξ )η +ξρ ξ + ηρ η = Transport equation for linear char. variable: W = V ξ U η = v ξ u η UW ξ + V W η + (U ξ + V η + 1)W = w = n ξ m η (ξ, η) w + w = or rm r = p ξ rn r = p η ξ

6 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 6 Converging Shocks: A Bifurcation Problem for NLWS y x= κ a y U 1 U x x=κ a y 2-state data: U, U 1 Data give 2 shocks Far field soln: 4 waves η U 1b Linear Wave Symmetric prototype for converging sector boundaries Weak shock reflection, von Neumann paradox Shock U 1 U ξ Shock U 1a Linear Wave 1. Parameterized by ρ /ρ 1 > 1 and by κ a 2. Incident shocks: ξ = κη χ, ξ = κη + χ 3. Linear waves: angle of incidence = angle of reflection 4. Small κ: two local solutions weak and strong regular reflection 5. Large κ: curved shock, weak reflected wave (cf. κ = ) 6. Intermediate values of κ: no solution from shock polar analysis

7 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 7 Converging Shock Data for the Nonlinear Wave System 15 Regions of Different Qualitative Behavior; γ = 2 x= κ a y y U =( ρ,, ) 1 1 x= κ y a x 1 κ * Region A: WMR U =(ρ,, n ) κ a ξ= κ a η ξ= κ a χ a + U U 1b l b U 1 U1a l a S b + S a C 1 U ξ=κ a +χ a ξ=κ a η U C 5 Region B: MR Region C: RR ρ /ρ Three regions: A Weak MR possible C RR possible B neither possible

8 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 8 Some Numerical Simulations of the Problem Alex Kurganov, Tulane University Godunov-type central scheme, central-upwind scheme, Alexander Kurganov, Sebastian Noelle and Guergana Petrova; modified version, Kurganov and Chi-Tien Lin. SISC, 23, 21, pp ( kurganov). Central schemes (eg. Lax-Friedrichs scheme): avoid solving RP by integrating over local Riemann fans (at cell interface). Second-order staggered Nessyahu-Tadmor scheme (199), later extended to higher orders and to multi-d. Kurganov & Tadmor, 1999: nonstaggered central schemes, with lower dissipation, simple semi-discrete form. New, central-upwind schemes: more accurate estimate of size of Riemann fans and more accurate projection step.

9 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 9 Regular Reflection: κ a =.5

10 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 1 Intermediate Angle, κ a = 1, Region B Contour Plot of m: U = (64,, ); U 1 = (1,,); κ a = 1; κ b = η axis ξ axis Contour Plot of n. U = (64,, ); U 1 = (1,,); κ a = 1; κ b = 1 Contour Plot of Density ρ. Data U = (64,, ); U 1 = (1,,); κ a = 1; κ b = η axis η axis ξ axis ξ axis

11 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, An Intermediate Angle, κ a = 2: Weak or Mach Reflection? Contour Plot of m: U = (64,, ); U 1 = (1,,); κ a = 2; κ b = η axis ξ axis Contour Plot of n. U = (64,, ); U 1 = (1,,); κ a = 2; κ b = 2 Contour Plot of Density ρ. Data U = (64,, ); U 1 = (1,,); κ a = 2; κ b = η axis η axis ξ axis ξ axis

12 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Weak Mach Reflection (Large κ a ): Region C Contour Plot of Density ρ. Data U = (64,, ); U 1 = (1,,); κ a = 8; κ b = η axis ξ axis Sonic circle C = {ξ 2 +η 2 = c 2 (ρ )} Supersonic soln known U continuous at C U/ r singular

13 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Weak Mach Reflection (Large κ a ): Momentum Component m

14 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Analysis of Weak Mach Stem Problem (Large κ) Degenerate Elliptic Free Boundary Problem Existence theorem for global problem for NLWS Q ( ) (c 2 (ρ) ξ 2 )ρ ξ ξηρ η + ( (c 2 (ρ) η 2 )ρ ξ η ξηρ ξ )η + ξρ ξ + ηρ η Q(ρ) = (degenerate elliptic) in Ω Σ c 2 (ρ) = ξ 2 + η 2 = c 2 (ρ ) on σ (degenerate boundary, continuous solution) ρ ξ = (symmetry) on Σ Free boundary from RH equations: Σ Ω σ N(ρ) β ρ = (oblique deriv) on Σ dη dξ = η 2 s 2 ξη + s 2 (ξ 2 + η 2 s 2 ) s 2 = [p] [ρ] ρ = ρ max at Σ Σ (part of D. bdry) Approach: Fixed Point Theorem (CK & Lieberman, CKK) New difficulties: N not uniformly oblique; est. at degenerate corner

15 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Previous Results on Related Free Boundary Problems Fixed point approach developed in joint work with Gary Lieberman (F B problem in steady TSD equation: C, K & Lieberman) Partial solution for regular reflection in UTSD model, (a > 2): 2 types of regular reflection weak & strong a = (1 + 5/2) 1/2 ( 2, a ): both subs a > a : 1 sub, 1 sup Riemann data: U = (, ), x > a y, U 1 = (1, a), y >, x < ay U 1 = (1, a), y <, x < ay Reflected Shock ELLIPTIC REGION WEAK Sonic Line FREE BOUNDARY Incident Shock Reflected Shock ELLIPTIC REGION STRONG DEGENERACY IN ELLIPTIC EQUATION Incident Shock Complete description of flow needs: asymptotic behavior at ξ = solving degen elliptic prob solving F B prob for shock Unbounded subsonic region: artifact of UTSD model

16 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Procedure: Find the Free Boundary as a Fixed Point Formulate as 2nd order PDE for density, ρ (not potential); Rewrite RH conditions as (1) evolution eqn for shock and (2) ODBC for ρ Problem is quasilinear, degenerate elliptic PDE, mixed BC Regularize PDE (parameter ε) Step 1 Fix approx. η = η(ξ), defines Σ K ε H 1+α1 (Hölder) Step 2 Solve (fixed) mixed BVP for ρ Lieberman s Mixed BVP theory + linearization + modifications for loss of obliqueness Step 3 Map η η = Jρ by other RH condition (shock evolution) Schauder F. P. Thm: Compactness fixed pt for J J : K H 1+α1 K H 1+α, α > α 1 Step 4 Show η and ρ solve the problem.

17 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Degenerate Elliptic Equations Principal part: ( (c 2 (ρ) U 2 )ρ ξ UV ρ η )ξ +( (c 2 (ρ) V 2 )ρ η UV ρ ξ ) with U = u ξ, V = v η or U = ξ, V = η Linear prototype: Tricomi, u xx + xu yy, or Keldysh, xu xx + u yy η ξ SUBSONIC REGION: ELLIPTIC OR MIXED DEGENERATE CHARACTERISTICS τ τ + NONDEGENERATE SPACELIKE CURVE Ξ CHARACTERISTICS SUPERSONIC REGION: HYPERBOLIC L Differ on Hyperb. side (x < ): dir. of char. Differ on Elllliptic side (x > ): Fichera fn b (b k a kj x j )n k (NB: l. o. terms) (Nonlinear definition ambiguous) Keldysh eqns permit Dirichlet BC (need in nonlin. eqn) Linear Keldysh solns have singularities at x = : u = x γ Nonlinear Keldysh solns are regular OR singular η

18 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, P 1 P 2 Mixed BC: Weighted Hölder Norms for Corner Singularities Ck,α = H k+α : u k+α = j<k Dj u + D k u α Ω Partially interior or weighted norms (corners) δ u (b) a;ω S = sup c δ δ a+b u a;ωδ, H (b) a;ω S c Example: S = V {P 1, P 2 }; u = r γ H ( γ) 1+α Lin Prob, z H ( γ 1) 1+ɛ : Lu = D i ( a ij (z)d j u ), Mu = β(z) u Estimates indep of z: u ( γ) 1+α C 1 ( f γ + u ) Nonlinear problem: Harnack ineq est with α indep of α 1 Schauder Fixed Point Theorem: B = H 1+α1 ; α 1 min { α 2, γ } 2 K closed, bdd., cvx. K B; J : K K, JK K J has fixed pt. γ dep. on corners only; α indep. of α 1 ; Jρ H 1+α, α > α 1

19 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, A Priori Estimates (following Lieberman, Zheng, Canic&Kim, CKL) Regularize: Q ε ρ = Qρ + ε ρ. Obliqueness fails at Ξ s. Step 1: for FBP for Q ε ; a priori bds on ρ ε, ρ ɛ (η) unif. in ɛ. Step 2: Local lower barrier for ρ ε independent of ε, unif. local ellipticity local compactness. Step 3: Convergent subsequence; limit solves the problem in Ω \ {Ξ }: applying regularity, compactness and a diagonalization. Step 4: Convergence at Ξ requires more. Ξ Ξ Σ η Σ Ω Ξ σ ξ

20 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, 23 2 The Upper Barrier at Ξ Barrier construction (supersolution): Qψ, Nψ, ψ ρ on bdry of Ω(a, h) Q ε ρ = (c 2 r 2 + ε)ρ rr + c2 r 2 ρ θθ +p (ρ 2 r + 1 r 2 ρ 2 θ ) + ( c2 r 2r)ρ r Upper barrier: ψ(r, θ) = ρ + A(c r) b + B(θ 1 θ) 2 A, B, b (, 1) to be determined from Q ε ψ = (c r) b 2 p B(θ 1 θ) 2 b(b 1)A+... Singular barriers in degenerate eqns: θ Choi-McKenna &Canic-Kim. 1 Conjecture: solution has sing. at C ; b ( < 1/2) optimal ( ) (c r) 2b 2 : A 2 b p (ρ)(b 1) + p (ρ)b A 2 k < if b < Also need ψ ρ ε on Ω(a, h). so A >> 1, Q ε ψ <. Σ Ξ σ Ω (a,h) min p 2 max p,

21 Multiphase Fluid Flows and Multi-Dimensional Hyperbolic Problems, Newton Institute, March 31-April 4, Completing the Upper Barrier at Ξ On σ: ρ ε = ρ < ψ. On {θ = θ 1 a}: Choose Ba 2 > ρ M. On {r = c h}: Choose A so Ah b > ρ M. Σ Ξ σ Ω (a,h) On Σ, ODBC: if N 1 (ρ ε )ψ β(ρ ε ) ψ, then ψ ρ ε >. Now ψ = (ψ r, ψ θ ) as r c : ψ n AND ψ t : usual barrier construction fails. But N 1 ψ = β r ψ r + β θ ψ θ, Now β 1 ξ + β 2 η (η η ξ)(ξ + η η) }{{} > by geometry β r = 1 r (β 1ξ + β 2 η), ( r 2 (c 2 (ρ) + 3s 2 (ρ, ρ 1 )) 4c 2 s 2). }{{} > at r=c,ρ=ρ A priori bound: need β 1 ξ + β 2 η > for all ρ (ρ, ρ M ). But ρ M = ρ + O(1/κ a ). If ρ >> ρ 1, then κ a 8 suffices.