Multidimensional Conservation Laws Part II: Multidimensional Models and Problems. BCAM and UPV/EHU Courses 2010/ PDF Free Download

Multidimensional Conservation Laws Part II: Multidimensional Models and Problems Gui-Qiang G. Chen Oxford Centre for Nonlinear PDE Mathematical Institute, University of Oxford Website: http://people.maths.ox.ac.uk/chengq/ BCAM and UPV/EHU Courses 2010/2011 Basque Center for Applied Mathematics Bizkaia Technology Park, Building 500 E-48160 Derio - Spain September 12 15, 2011 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 1 / 84

Basic References I Gui-Qiang Chen, Multidimensional Conservation Laws: Overview, Problems, and Perspective, In: Nonlinear Conservation Laws and Applications, IMA Volume 153 in Mathematics and Its Applications, pp. 23 72, Eds. A. Bressan, G.-Q. Chen, M. Lewicka, and D. Wang, Springer-Verlag: New York, 2010 Gui-Qiang Chen, Euler Equations and Related Hyperbolic Conservation Laws, In: Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 2, pp. 1-104, 2005, Elsevier: Amsterdam, Netherlands Gui-Qiang Chen and Mikhail Feldman, Shock Reflection-Diffraction and Multidimensional Conservation Laws, In: Hyperbolic Problems: Theory, Numerics and Applications, pp, 25 51, Proc. Sympos. Appl. Math. 67, Part 1, AMS: Providence, RI, 2009. Gui-Qiang Chen, Monica Torres, and William Ziemer, Measure-Theoretical Analysis and Nonlinear Conservation Laws, Pure Appl. Math. Quarterly, 3 (2007), 841 879. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 2 / 84

Basic References II Gui-Qiang Chen and Mikhail Feldman, Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow, Annals of Mathematics, 171 (2010), 1067 1182. Gui-Qiang Chen and Ya-Guang Wang, Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws, Proceedings of the Abel Symposium on Nonlinear PDE, Springer, 2011 (to appear) Gui-Qiang Chen, Monica Torres, and William Ziemer, Gauss-Green Theorem for Weakly Differentiable Fields, Sets of Finite Perimeter, and Balance Laws, Communications on Pure and Applied Mathematics, 62 (2009), 242 304. Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang, Conservation Laws: Transonic Flow and Differential Geometry, Proc. Sympos. Appl. Math. 67, Part 1, pp. 217 226, AMS: Providence, RI, 2009. Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Third edition. Springer-Verlag: Berlin, 2010. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 3 / 84

Hyperbolic Conservation Laws t u + f(u) = 0 u = (u 1,, u m ), x = (x 1,, x d ), = ( x1,, xd ) f = (f 1,, f d ) : R m (R m ) d is a nonlinear mapping f i : R m R m for i = 1,, d t A(u, u t, u) + B(u, u t, u) = 0 A, B : R m R m (R m ) d R m are nonlinear mappings Connections and Applications: Fluid Mechanics and Related: Euler Equations and Related Equations Gas, shallow water, elastic body, reacting gas, plasma,... Special Relativity: Relativistic Euler Equations and Related Equations General Relativity: Einstein Equations and Related Equations Differential Geometry: Isometric Embeddings, Nonsmooth Manifolds.. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 4 / 84

Approaches and Strategies: Proposal Diverse Approaches in Sciences: Experimental data Large and small scale computing by a search for effective numerical methods Modelling (Asymptotic and Qualitative) Rigorous proofs for prototype problems and an understanding of the solutions Two Strategies as a first step: Study good, simpler nonlinear models with physical motivations; Study special, concrete nonlinear problems with physical motivations Meanwhile, extend the results and ideas to: Study the Euler equations in gas dynamics and elasticity Study nonlinear systems that the Euler equations are the main subsystem or describe the dynamics of macroscopic variables such as MHD, Euler-Poisson Equations, Combustion, Relativistic Euler Equations, Study more general hyperbolic systems and related problems Develop further new mathematical ideas, techniques, approaches, as well as new mathematical theories Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 5 / 84

Outline 1 Important Multidimensional Models 2 Multidimensional Steady Problems 3 Multidimensional Self-Similar Problems Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 6 / 84

Unsteady Transonic Small Disturbance Equation (E-9) UTSD equation (E-9) in transonic aerodynamics: { t u + x ( 1 2 u2 ) + y v = 0, x v y u = 0, or the Zabolotskaya-Khokhlov equation (E-10): x ( t u + u x u) + yy u = 0. (E-9) describes the potential flow field near the reflection point in weak shock reflection, which determine the leading order approximation of geometric optical expansions. It can be also used to formulate asymptotic equations for the transition from regular to Mach reflection for weak shocks. It also describes high-frequency waves near singular rays (Hunter 1986). (E-10) was first derived by Timman (1964) in the context of transonic flows and by Zabolotskaya-Khokhlov (1969) in nonlinear acoustics which describes the diffraction of nonlinear acoustic beams. Cramer-Seebass (1978) used (E-7) to study caustics in nearly planar sound waves. The same equation arises as a weakly nonlinear equation for cusped caustics. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 7 / 84

Steady Transonic Small Disturbance Equation (E-11) or x (u x u) + yy u = 0 u xx u + yy u + ( x u) 2 = 0. Elliptic: u > 0 Hyperbolic: u < 0 This is a nonlinear version of the celebrated linear equations of mixed type: Tricomi Equation: xx u + x yy u = 0 (hyperbolic degeneracy at x = 0) Keldysh Equation: x xx u + yy u = 0 (parabolic degeneracy at x = 0) J. Hunter, C. Morawetz, B. Keyfitz, S. Canic, G. Lieberman, Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 8 / 84

Pressureless Euler Equations (E-12) Separating the pressure from the Euler equations in R 2, motivated by the flux-splitting scheme used by Argarwal-Halt (1994) in numerical computations for airfoil flows. It may be obtained from the infinite Mach number limit from the full Euler equations. The pressureless Euler equations, modelling the motion of free particles which stick under collision, (E-12): t ρ + x (ρu) + y (ρv) = 0, t (ρu) + x (ρu 2 ) + y (ρuv) = 0, t (ρv) + x (ρuv) + y (ρv 2 ) = 0, t (ρe) + x (ρue) + y (ρve) = 0. All the characteristic families are linear degenerate; but solutions may become measure solutions due to concentration under collision. Zeldovich 1970 Bouchut 1994, Grenier 1995, E-Rykov-Sinai 1996, Wang-Huang-Ding 1997 Brenier-Grenier 1998, Bouchut-James 1999, Huang-Wang 2001, Li-Zhang 1998-99, Huang-Yang 1998, Sheng-Zhang 1999, Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 9 / 84

Pressure-Gradient System (E-13) t ρ = 0, t (ρu) + x p = 0, t (ρv) + y p = 0, t (ρe) + x (up) + y (vp) = 0. For small velocity and large gas constant γ, ρe = 1 2 ρ(u2 + v 2 ) + 1 γ 1 p is 1 1 dominated by γ 1p, then, by setting p = (γ 1)P, t = γ 1τ, we have τ u + x P = 0, τ v + y P = 0, τ P + P x u + P y v = 0. Eliminating (u, v), we obtain the nonlinear wave equations (E-13): Yuxi Zheng, Kyungwoo Song, τ ( 1 P τ P) P = 0. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 10 / 84

Nonlinear Wave System (E-14) t ρ + x (ρu) + y (ρv) = 0, t (ρu) + x (ρu 2 + p) + y (ρuv) = 0, t (ρv) + x (ρuv) + y (ρuv + p) = 0, For small velocity and irrotational flow, ignore the nonlinear velocity terms and denote (m, n) = (ρu, ρv) as momenta: Eliminating (m, n), we have Suny Canic, Barbara Keyfitz, t ρ + x m + y n = 0, t m + x p = 0, t n + y p = 0. tt ρ p(ρ) = 0 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 11 / 84

Euler Equations for the Chaplygin Gas (E-15) { t ρ + (ρv) = 0, v R d, x R d t (ρv) + (ρv v ) + p(ρ) = 0. For the Chaplygin gas, also called the von Karmen gas, p(ρ) = 1 ρ, Advantage: The pressure waves are contact discontinuities, and their location is often known apriori, instead of being a free boundary. Drawback: Since the pressure is uniformly bounded, some Riemann problems yield a concentration of mass along a co-dimension one subset. Serre 2009: Multidimensional shock waves and Riemann problems Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 12 / 84

Gauss-Codazzi Equations in the Fluid Dynamics Formalism for Isometric Embedding (E-16) The Codazzi Equations (i.e. the Balance of Momentum Equations): { x (ρuv) + y (ρv 2 + p) = Γ (2) 22 (ρv 2 + p) 2Γ (2) 12 ρuv Γ(2) 11 (ρu2 + p), x (ρu 2 + p) + y (ρuv) = Γ (1) 22 (ρv 2 + p) 2Γ (1) 12 ρuv Γ(1) 11 (ρu2 + p), The Gauss Equation (i.e. the Bernoulli Relation): p = q 2 + K, q 2 = u 2 + v 2, K Gaussian curvature Constitutive relation the Chaplygin type gas: p(ρ) = 1 ρ Γ (k) ij Christoffel symbols, depending on the metric g ij up to their 1st derivatives, i, j, k = 1, 2. Define the sound speed: c 2 = p (ρ). Then c 2 = 1/ρ 2 = q 2 + K. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 13 / 84

Gauss-Codazzi Equations in the Fluid Dynamics Formalism for Isometric Embedding (E-16) The Codazzi Equations (i.e. the Balance of Momentum Equations): { x (ρuv) + y (ρv 2 + p) = Γ (2) 22 (ρv 2 + p) 2Γ (2) 12 ρuv Γ(2) 11 (ρu2 + p), x (ρu 2 + p) + y (ρuv) = Γ (1) 22 (ρv 2 + p) 2Γ (1) 12 ρuv Γ(1) 11 (ρu2 + p), The Gauss Equation (i.e. the Bernoulli Relation): p = q 2 + K, q 2 = u 2 + v 2, K Gaussian curvature Constitutive relation the Chaplygin type gas: p(ρ) = 1 ρ Γ (k) ij Christoffel symbols, depending on the metric g ij up to their 1st derivatives, i, j, k = 1, 2. Define the sound speed: c 2 = p (ρ). Then c 2 = 1/ρ 2 = q 2 + K. c 2 > q 2 and the flow is subsonic when K > 0, c 2 < q 2 and the flow is supersonic when K < 0, c 2 = q 2 and the flow is sonic when K = 0. G.-Q. Chen, M. Slemrod, D.-H. Wang: CMP 2010 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 13 / 84

The Gaussian Curvature K on a Torus: Doughnut Surface or Toroidal Shell K<0 K>0 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 14 / 84

Lax System: Complex Inviscid Burgers Equation (E-17) f (u) Complex valued function of a single complex variable u = u + vi u = u(t, z) Complex valued function with z = x + yi and t R Lax System: t ū + z f (u) = 0, z = 1 2 ( x i y ) For u = u + iv and 1 2f (u) = a(u, v) + b(u, v)i: { t u + x a(u, v) + y b(u, v) = 0, t v x b(u, v) + y a(u, v) = 0. When f (u) = u 2 = u 2 + v 2 + 2uvi, the complex Burger equation: { t u + 1 2 x(u 2 + v 2 ) + y (uv) = 0, t v x (uv) + 1 2 y (u 2 + v 2 ) = 0. This is a symmetric hyperbolic system with an entropy η(u, v) = u 2 + v 2, so that local well-posedness of classical solutions can be inferred directly. For the 1-D case, this is an archetype of hyperbolic conservation laws with umbilic degeneracy: Schaeffer-Shearer: 1976, Chen-Kan: 1995, 2001, Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 15 / 84

Two-D Steady Euler Equations (E-18): (x, y) R 2 Constitutive Relations: x (ρu) + y (ρv) = 0, x (ρu 2 + p) + y (ρuv) = 0, x (ρuv) + y (ρv 2 + p) = 0, x ( ρu(e + p ρ )) + y ( ρv(e + p ρ )) = 0, (e, p, θ) = (e(ρ, S), p(ρ, S), θ(ρ, S)), E = e + 1 2 (u2 + v 2 ) ρ fluid density (u, v) fluid velocity p scalar pressure S entropy e internal energy θ temperature For a polytropic gas, p = Rρθ, e = p (γ 1)ρ, γ = 1 + R c v p = p(ρ, S) = κρ γ e S/cv, e = κ γ 1 ργ 1 e S/cv R > 0, c v > 0, κ > 0 are constants, γ > 1 is the adiabatic exponent Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 17 / 84

2-D Steady Full Euler Equations: Eigenvalues and Waves Eigenvalues for the 1 st and 4 th families: = λ j = uv + ( 1)j c u 2 + v 2 c 2 u 2 c 2, j = 1, 4. Supersonic when u 2 + v 2 > c 2 : Subsonic when u 2 + v 2 < c 2 : Shock waves, Rarefaction waves Elliptic equations Eigenvalues for the 2 nd and 3 rd families: λ i = v/u, i = 2, 3 = Two transport equations The 2 nd families: Vortex sheets The 3 rd families: Entropy waves *New Phenomena: Compressible vortex sheets Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 18 / 84

Steady Potential Flow Equation (E-5) (ρ( Φ) Φ) = 0 For a γ-law gas, p = p(ρ) = ρ γ /γ, γ > 1, is the normalized pressure. Then the normalized Bernoulli s law: Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 19 / 84

Steady Potential Flow Equation (E-5) (ρ( Φ) Φ) = 0 For a γ-law gas, p = p(ρ) = ρ γ /γ, γ > 1, is the normalized pressure. Then the normalized Bernoulli s law: ρ = ˆρ(q 2 ) := ( 1 γ 1 q 2) 1 γ 1, 2 Define c = 1 γ 1 2 q2 (sonic speed), q cr : We rewrite Bernoulli s law in the form q = u 2 + v 2 flow speed. 2 γ+1 q 2 qcr 2 = 2 ( q 2 c 2). γ + 1 (critical speed). Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 19 / 84

Steady Potential Flow Equation (E-5) (ρ( Φ) Φ) = 0 For a γ-law gas, p = p(ρ) = ρ γ /γ, γ > 1, is the normalized pressure. Then the normalized Bernoulli s law: ρ = ˆρ(q 2 ) := ( 1 γ 1 q 2) 1 γ 1, 2 Define c = 1 γ 1 2 q2 (sonic speed), q cr : We rewrite Bernoulli s law in the form q = u 2 + v 2 flow speed. 2 γ+1 q 2 qcr 2 = 2 ( q 2 c 2). γ + 1 Then the flow is subsonic (elliptic) when q < q cr, sonic (degenerate state) when q = q cr, supersonic (hyperbolic) when q > q cr. (critical speed). Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 19 / 84

Airfoil Problems I: (ρ( Φ) Φ) = 0, x R 2, v = 0 Obstacle Boundary Ω 1 : Solid curve in (a); Solid closed curve in (b). Far-field Boundary Ω 2 : Dashed line segments in both (a) and (b). Domain Ω: bounded by Ω 1 and Ω 2. Boundary conditions on the obstacle Ω: { Φ n = 0 on Ω 1, Consistent far-field boundary conditions on Ω 2, where n is the unit normal pointing into the flow region on Ω. In case (b), the circulation about the boundary Ω 2 is zero. Ω2 Ω2 Ω Ω Ω1 Ω1 (a) (b) Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 20 / 84

Airfoil Problems II: (ρ( Φ) Φ) = 0, x R 2, v = 0 Problem 1: Existence of global entropy solutions Yes: When the far-field velocity (u, 0) with u q for some q < q cr : There exists a global subsonic (not necessarily strictly subsonic) flow Bers-Shiffman 1958,, Chen-Dafermos-Slemrod-Wang 2007 Open: Existence of global transonic entropy solutions when u (q, q cr )? References: Morawetz: CPAM 38 (1985), 797 817; MAA 2 (1995), 257 268 Chen-Slemrod-Wang: ARMA 189 (2008), 159 188 A vanishing viscosity method Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 21 / 84

Wedge/Cone Problems I: Wedge Problems = Progress Wedge Problems u 2 T S O θ c H q u 1 Sonic Circle Supersonic-Supersonic: Smooth supersonic: Gu, Schaeffer, Li, S. Chen Discontinuous supersonic: Y. Zhang, Chen-Zhang-Zhu, Chen-Li, Supersonic-Subsonic: Fang, Chen-Fang, Chen-J. Chen-Feldman... Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 22 / 84

Wedge/Cone Problems II: Cone Problems Supersonic-Supersonic: Smooth supersonic: Chen-Xin-Yin, Discontinuous supersonic: Lien-T.-P. Liu, Chen-Zhang-Zhu Supersonic-Subsonic: Chen-Fang, Yin et al, Require: The incoming flow is sufficiently supersonic: In progress... Problem 2: Existence and stability of shock-fronts as long as the incoming flow is supersonic Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 23 / 84

Transonic Nozzle Problems Problem 3. Existence of subsonic-supersonic-subsonic solutions past a transonic nozzle The problem involves two types of transonic flow with two free boundaries: Free boundary from the subsonic to supersonic flow through a continuous transition; Free boundary from the supersonic to subsonic flow through a transonic shock. Existence and stability of supersonic-subsonic solutions: Chen-Feldman: 2003, 2004, 2007; Bae-Feldman 2009; Chen-Yuan; Xin-Yin,... * A. G. Kuz min: Boundary Value Problems for Transonic Flow, John Wiley & Sons LTD., 2002 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 24 / 84

Stability Problems for Steady Compressible Vortex Sheets Problem 4. Stability of Compressible Vortex Sheets in Two-Dimensional Steady Supersonic Euler Flows Chen-Zhang-Zhu (SIAM 2007), Chen-Kukreja (2011): M ± > 1: L 1 -Stability M. Bae (Preprint 2011): M ± < 1: Structural Stability Question: Stability of two or multiple steady compressible vortex sheets? Backgrounds: Motion and structure of galactic jets in astrophysics... Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 25 / 84

Two-Dimensional Steady Euler Equations: (x, y) R 2 Constitutive Relations: x (ρu) + y (ρv) = 0, x (ρu 2 + p) + y (ρuv) = 0, x (ρuv) + y (ρv 2 + p) = 0, x ( ρu(e + p ρ )) + y ( ρv(e + p ρ )) = 0, (e, p, θ) = (e(ρ, S), p(ρ, S), θ(ρ, S)), E = e + 1 2 (u2 + v 2 ) ρ fluid density (u, v) fluid velocity p scalar pressure S entropy e internal energy θ temperature For a polytropic gas, p = Rρθ, e = p (γ 1)ρ, γ = 1 + R c v p = p(ρ, S) = κρ γ e S/cv, e = κ γ 1 ργ 1 e S/cv R > 0, c v > 0, κ > 0 are constants, γ > 1 is the adiabatic exponent Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 26 / 84

Compressible Vortex Sheets/Entropy Waves The 2 nd vortex sheet and 3 rd entropy wave y = χ i (x): dy dx = λ i = v/u, i = 2, 3 The 2 nd -family vortex sheet curves in the phase space: C 2 (U 0 ) : U = (u 0 e σ 2, v 0 e σ 2, p 0, ρ 0 ), S = S 0, with strength σ 2 and slope v 0 /u 0, determined via the eigenvector r 2 by du dσ 2 = r 2 (U) = (u, v, 0, 0), U σ2 =0 = U 0. The 3 rd -family entropy wave curves in the phase space: C 3 (U 0 ) : U = (u 0, v 0, p 0, ρ 0 e σ 3 ), S = S 0 c v γσ 3, with strength σ 3 and slope v 0 /u 0, determined via the eigenvector r 3 by du dσ 3 = r 3 (U) = (0, 0, 0, ρ), U σ3 =0 = U 0. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 27 / 84

Shock Waves and Rarefaction Waves Eigenvalues for the 1 st and 4 th families: λ j = uv + ( 1)j c u 2 + v 2 c 2 u 2 c 2, j = 1, 4. The j th -family rarefaction wave curves in the phase space: R j (U 0 ) : dp = c 2 dρ, du = λ j dv, ρ(λ j u v)dv = dp for ρ < ρ 0, u > c. The j th -family shock wave curves in the phase space: S j (U 0 ) : [p] = c2 0 b [ρ], [u] = s j[v], ρ 0 (s j u 0 v 0 )[v] = [p] s j = u 0v 0 + ( 1) j c u0 2 + v 0 2 c2 u0 2 for ρ > ρ 0, u > c c2 where c 2 = c2 0 ρ b ρ 0, b = γ+1 2 γ 1 ρ 2 ρ 0, and ρ 0 < ρ is equivalent to the entropy condition on the shock wave. S j (U 0 ) contacts with R j (U 0 ) at U 0 up to second-order. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 28 / 84

Stability Problem in the Region over the Lipschitz Wall Two-Phase Free Boundary Problems Free Boundary: Γ := {y = χ(x) : x > 0}, χ (x) = v(x,χ(x)) u(x,χ(x)) Free Bdry Conditions: [p] = 0, [ v u ] = 0; [ρ] = 0 or [u] = [v] = 0 *Unlike the shock case, the free boundary Γ is now a characteristic surface with the characteristic boundary conditions. 0 * Vortex sheet Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 29 / 84

Stability Problem: Initial-Boundary Value Type The vortex sheet/entropy wave problem can be formulated into the following IBVP: Cauchy Condition: Boundary Condition: { U1, 0 < y < y U x=0 = 0, U 2, y > y0 (u, v) n = 0 on the wedge boundary {y = g(x)}. (2) (1) 0 * Vortex sheet Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 30 / 84

Lateral Riemann Solutions shock Rarefaction wave S 4 S 4 R 4 R 4 Figure: Wave curves in the (u, v)-plane for the lateral Riemann problem Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 31 / 84

Existence and Stability (Chen-Zhang-Zhu: SIMA 2007) There exist ε 0 > 0 and C > 0 such that, if TV (g ( )) < ε for ε ε 0, then there exists a pair of functions U BV loc (Ω) L (Ω), χ Lip(R + ; R + ) with χ(0) = y0 such that U is a global entropy solution of the IBVP in Ω satisfying TV {U(x, ) : [g(x), )} C TV (g ( )) for every x [0, ); The curve {y = χ(x)} is a strong vortex sheet/entropy wave with χ(x) > g(x) for any x > 0, sup U(x, y) U 1 Cε, g(x)<y<χ(x) sup U(x, y) U 2 Cε, y>χ(x) lim sup { v(x, y) x u(x, y) g } : y > g(x) + lim x χ (x) g = 0; There exists a constant p > 0 such that lim x sup{ p(x, y) p : y > g(x)} = 0. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 32 / 84

Estimates on Weak Wave Interactions: Classical {U b, U m } = (α 1, α 2, α 3, α 4 ), {U m, U a } = (β 1, β 2, β 3, β 4 ), {U b, U a } = (γ 1, γ 2, γ 3, γ 4 ) Then γ i = α i + β i + O(1) (α, β), (α, β) = ( α 4 + α 3 + α 2 ) θ 1 + α 4 ( θ 2 + θ 3 ) + j=1,4 j(α, θ) { 0, αj 0, θ with j (α, θ) = j 0; α j θ j, otherwise. U a U a U m U b Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 33 / 84 U b

Estimates on the Weak Wave Reflections on the Boundary {U m, U a } = ( θ 1, θ 2, θ 3, 0), {U k, U m } = (0, 0, 0, α 4 ) with (u k, v k ) n k = 0. Then 1 U k+1 s.t. {U k+1, U a } = (0, 0, 0, δ 4 ), U k+1 (n k+1, 0, 0) = 0, and δ 4 = α 4 + K b1 θ1 + K b2 θ2 + K b3 θ3 + K b0 ω k, where K bj, j = 0, 1, 2, 3, are C 2 functions of θ 3, θ 2, θ 1, α 4, ω k+1, U a with K b1 {ωk =α 4 = θ 1 = θ 2 = θ 3 =0,U a=u 1 } = 1, K bi {ωk =α 4 = θ 1 =α 2 = θ 3 =0,U a=u 1 } = 0, i = 2, 3, and K b0 is bounded. In particular, K b0 < 0 at the origin. U a U a U m U k U k+1 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 34 / 84

Weak Waves Approach the Strong Vortex Sheet/Entropy Wave from Below: Essential Feature {U b, U m } = (0, α 2, α 3, α 4 ), {U m, U a } = (β 1, σ 2, σ 3, 0), {U b, U a } = (δ 1, σ 2, σ 3, δ 4 ) Then δ 1 = β 1 + K 11 α 4 + O(1), δ 4 = K 14 α 4 + O(1), σ 2 = σ 2+α 2 +K 12 α 4 +O(1), σ 3 = σ 3+α 3 +K 13 α 4 +O(1), with K 11 {α4 =α 3 =α 2 =0,σ 2 =σ 20,σ 3 =σ 30} < 1, and 4 j=2 K 1j is bounded, where = β 1 ( α 2 + α 3 ). U a U a Um U b U b Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 35 / 84

Weak Waves Approach the Strong Vortex Sheet/Entropy Wave from Above {U b, U m } = (0, σ 2, σ 3, α 4 ), {U m, U a } = ( θ 1, θ 2, θ 3, 0) Then {U b, U a } = (δ 1, σ 2, σ 3, δ 4 ). δ 1 = K 21 θ 1 + O(1), σ 2 = σ 2 + θ 2 + K 22 θ 1 + O(1), σ 3 = σ 3 + θ 3 + K 23 θ 1 + O(1), δ 4 = α 4 + K 24 θ 1 + O(1), with 4 j=1 K 2j is bounded, where = α 4 ( θ 2 + θ 3 ). U a U a Um U b Ub Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 36 / 84

Glimm-type Functional F (J) on the Mesh Curves J with F (J) = C ( σ J 2 σ 20 + σ J 3 σ 30 ) + L 1 (J) + L 2 (J) + KQ(J), L 1 (J) = K 1 L 0(J) + K 11 L1 1 (J) + K 12 L1 2 (J) + K 13 L1 3 (J) + L1 4 (J), L 2 (J) = K 21 L2 1 (J) + K 22 L2 2 (J) + K 23 L2 3 (J) + K 24 L2 4 (J), Q(J) = { α i β j : both α i and β j cross J and approach}, L 0 (J) = { ω(c k ) : C k the corner points in J + and the boundary}, L i j (J) = { α j : α j crosses J in region (i)}, i = 1, 2, j = 1, 2, 3, 4, (σ2 J, σj 3 ) Strength of the strong vortex sheet/entropy wave crossing J, where K and C will be chosen and K 1 > K b0, K 1j > K bj, j = 1, 2, 3, K24 < 1 K 11 K11, K21 > K 21 K11 + K 24 K K 14 24, while K12, K 13, K 22, and K 23 are arbitrarily large positive constants. These conditions can be achieved from our discussions of the properties of K bj, K Gui-Qiang, and K Chen, (Oxford) j = 1, 2, 3, 4. Multidimensional Conservation Laws September 12 16, 2011 37 / 84

Near the Strong Vortex Sheet/Entropy Waves (I) (L 1 + L 2 )(J) (L 1 + L 2 )L(I ) ( K 14 K14 + K 11 K11 1) α 4 (K12 +O(1) θ 1 ) α 2 (K13 +O(1) θ 1 ) α 3 with K 14 K14 + K 11 K11 1 < 0 by our choice of the constants. Furthermore, since Q(I ) can always be bounded by L(I ) and σ2 J σi 2 + σj 3 σi 3 ( K 12 + K 13 ) α 4 + (1 + O(1) θ 1 ) α 2 + (1 + O(1) θ 1 ) α 3 with K 12 and K 13 bounded, we can choose C suitably small such that F (J) F (I ) C ( σ2 J σi 2 + σj 3 σi 3 ) +(L 1 (J) + L 2 (J) + KQ(J)) (L 1 (I ) + L 2 (I ) + KQ(I )) 0. (1) Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 38 / 84

Near the Strong Vortex Sheet/Entropy Waves (II) (L 1 + L 2 )(J) (L 1 + L 2 )(I ) ( K 21 K 11 + K 24 K 24 K 21 ) θ 1 (K 22 + O(1) α 4 ) θ 2 (K 23 + O(1) α 4 ) θ 3 with K 21 K 11 + K 24 K 24 K 21 < 0 by our choice of the constants. Similar to the analysis for Case (I), we again have F (J) F (I ). (2) Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 39 / 84

Remarks and Connections Glimm schemes, wave-front tracking schemes... Contact discontinuities for 1-D strictly hyperbolic systems: Corli and Sablé-Tougeron (1997), Sablé-Tougeron (1993),?? Connections between the stability of steady compressible vortex sheets/entropy waves and long-time asymptotic stability of unsteady compressible vortex sheets/entropy waves in supersonic flow Three-dimensional, steady vortex sheets/entropy waves?? Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 40 / 84

? Shock Wave Patterns around a Wedge (airfoils, inclined ramps, ) Complexity of Reflection-Diffraction Configurations Was First Identified and Reported by Ernst Mach 1879 Experimental Analysis: 1940s= von Neumann, Bleakney, Bazhenova Glass, Takyama, Henderson, Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 42 / 84 Shock Reflection-Diffraction Problems

Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 43 / 84

Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 44 / 84

Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 45 / 84

0.4108 0.4106 Subsonic 0.4104 y/t 0.4102 Supersonic 0.41 0.4098 1.0746 1.0748 1.075 1.0752 1.0754 1.0756 x/t A New Mach Reflection-Diffraction Pattern: A. M. Tesdall and J. K. Hunter: TSD, 2002 A. M. Tesdall, R. Sanders, and B. L. Keyfitz: NWE, 2006; Full Euler, 2008 B. Skews and J. Ashworth: J. Fluid Mech. 542 (2005), 105-114 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 46 / 84

Shock Reflection-Diffraction Patterns Gabi Ben-Dor Shock Wave Reflection Phenomena Springer-Verlag: New York, 307 pages, 1992. Experimental results before 1991 Various proposals for transition criteria Milton Van Dyke An Album of Fluid Motion The parabolic Press: Stanford, 176 pages, 1982. Various photographs of shock wave reflection phenomena Journals by Springer: Shock Waves Combustion, Explosion, and Shock Waves Richard Courant & Kurt Otto Friedrichs Supersonic Flow and Shock Waves Springer-Verlag: New York, 1948. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 47 / 84

Scientific Issues Structure of the Shock Reflection-Diffraction Patterns Transition Criteria among the Patterns Dependence of the Patterns on the Parameters wedge angle θ w, adiabatic exponent γ 1 incident-shock-wave Mach number M s Interdisciplinary Approaches: Experimental Data and Photographs Large or Small Scale Computing Colella, Berger, Deschambault, Glass, Glaz, Woodward,... Anderson, Hindman, Kutler, Schneyer, Shankar,... Yu. Dem yanov, Panasenko,... Asymptotic Analysis Lighthill, Keller, Majda, Hunter, Rosales, Tabak, Gamba, Harabetian... Morawetz: CPAM 1994 Rigorous Mathematical Analysis?? (Global Solutions) Existence, Stability, Regularity, Bifurcation, Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 48 / 84

2-D Riemann Problem for Hyperbolic Conservation Laws t U + x F (U) = 0, x = (x 1, x 2 ) R 2 t=0 U U 3 0 N-1 U U 2 N U 1 Books and Survey Articles: Chang-Hsiao 1989, Glimm-Majda 1991, Li-Zhang-Yang 1998, Zheng 2001 Chen-Wang 2002, Serre 2005, Chen 2005, Dafermos 2010, Numerical Solutions: Glimm-Klingenberg-McBryan-Plohr-Sharp-Yaniv 1985 Lax-Liu 1998, Schulz-Rinne-Collins-Glaz 1993, Chang-Chen-Yang 1995, 2000 Kurganov-Tadmor 2002, Theoretical Roles: Asymptotic States and Attractors Local Structure and Building Blocks... Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 49 / 84

Riemann Solutions I Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 50 / 84

Riemann Solutions II Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 51 / 84

Riemann Solutions vs General Entropy Solutions Asymptotic States and Attractors Observation (C Frid 1998): Let R( x t ) be the unique piecewise Lipschitz continuous Riemann solution with Riemann data: R t=0 = R 0 ( x x ) Let U(t, x) be a bounded entropy solution with initial data: U t=0 = R 0 ( x x )+P 0(x), R 0 L (S d 1 ), P 0 L 1 L (R d ) The corresponding self-similar sequence U T (t, x) := U(Tt, T x) is compact in L 1 loc (Rd+1 + ) = ess lim U(t, tξ) R(ξ) dξ = 0 t Ω Building Blocks and Local Structure Local structure of entropy solutions Building blocks for numerical methods for any Ω R d Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 52 / 84

Full Euler Equations (E-1): (t, x) R 3 + := (0, ) R 2 t ρ + (ρv) = 0 t (ρv) + (ρv v) + p = 0 t ( 1 2 ρ v 2 + ρe) + (( 1 2 ρ v 2 + ρe + p)v ) = 0 Constitutive Relations: p = p(ρ, e) ρ density, v = (v 1, v 2 ) fluid velocity, p pressure e internal energy, θ temperature, S entropy For a polytropic gas: p = (γ 1)ρe, e = c v θ, γ = 1 + R c v p = p(ρ, S) = κρ γ κ e S/cv, e = e(ρ, S) γ 1 ργ 1 e S/cv, R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas c v > 0 is the specific heat at constant volume γ > 1 is the adiabatic exponent, κ > 0 is any constant under scaling Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 53 / 84

Euler Equations for Potential Flow: (E-3): v = Φ { t ρ + (ρ Φ) = 0, (conservation of mass) or, equivalently, t Φ + 1 2 Φ 2 + ργ 1 γ 1 = ργ 1 0 γ 1, (Bernoulli s law); t ρ( t Φ, Φ, ρ 0 ) + (ρ( t Φ, Φ, ρ 0 ) Φ ) = 0, with ρ( t Φ, Φ, ρ 0 ) = ( ρ γ 1 0 (γ 1)( t Φ + 1 2 Φ 2 ) ) 1 γ 1. We will see the Euler equations for potential flow is actually EXACT in an important region of the solution to the shock reflection problem. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 54 / 84

Discontinuities of Solutions t u + f(u) = 0, x = (x 1,, x d ) R d An oriented surface Γ(t) with unit normal n = (n t,, n d ) R d in the (t, x)-space is a discontinuity of a piecewise smooth entropy solution U with { u + (t, x), (t, x) n > 0, u(t, x) = u (t, x), (t, x) n < 0, if the Rankine-Hugoniot Condition is satisfied (u + u, f(u + ) f(u )) n = 0 along Γ(t). The surface (Γ(t), u) is called a Shock Wave if the Entropy Condition (i.e., the Second Law of Thermodynamics) is satisfied: (η(u + ) η(u ), q(u + ) q(u )) n > 0 along Γ(t), for some (η(u), q(u)): 2 η(u) 0, q j (u) = η(u)f j (u), j = 1,, d Example: For the full Euler equations: (η(u), q(u)) = ( ρs, ρvs). Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 55 / 84

Two Types of Discontinuities Noncharacteristic Discontinuities: Shock Waves: Characteristic Discontinuities: Vortex Sheets/Entropy Waves Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 56 / 84

Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 57 / 84

Initial-Boundary Value Problem: 0 < ρ 0 < ρ 1, v 1 > 0 Initial condition at t = 0: { (0, 0, p 0, ρ 0 ), x 2 > x 1 tan θ w, x 1 > 0, (v, p, ρ) = (v 1, 0, p 1, ρ 1 ), x 1 < 0; Slip boundary condition on the wedge bdry: v ν = 0. X2 (1) (0) Shock X1 Invariant under the Self-Similar Scaling: (t, x) (αt, αx), α 0 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 58 / 84

Seek Self-Similar Solutions (v, p, ρ)(t, x) = (v, p, ρ)(ξ, η), (ξ, η) = ( x 1 t, x 2 t ) (ρu) ξ + (ρv ) η + 2ρ = 0, (ρu 2 + p) ξ + (ρuv ) η + 3ρU = 0, (ρuv ) ξ + (ρv 2 + p) η + 3ρV = 0, (U( 1 2 ρq2 + γp γ 1 )) ξ + (V ( 1 2 ρq2 + γp γ 1 )) η + 2( 1 2 ρq2 + γp γ 1 ) = 0, where q = U 2 + V 2 and (U, V ) = (v 1 ξ, v 2 η) is the pseudo-velocity. Eigenvalues: λ 0 = V U (repeated), λ ± = UV ±c q 2 c 2 U 2 c 2, where c = γp/ρ is the sonic speed When the flow is pseudo-subsonic: q < c, the system is hyperbolic-elliptic composite-mixed Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 59 / 84

Boundary Value Problem in the Unbounded Domain Slip boundary condition on the wedge boundary: (U, V ) ν = 0 on D Asymptotic boundary condition as ξ 2 + η 2 : { (0, 0, p 0, ρ 0 ), ξ > ξ 0, η > ξ tan θ w, (U + ξ, V + η, p, ρ) (v 1, 0, p 1, ρ 1 ), ξ < ξ 0, η > 0. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 60 / 84

Normal Reflection When θ w = π/2, the reflection becomes the normal reflection, for which the incident shock normally reflects and the reflected shock is also a plane. reflected shock (1) (2) location of incident shock sonic circle elliptic sonic circle hyperbolic Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 61 / 84

von Neumann Criteria & Conjectures (1943) Local Theory for Regular Reflection (cf. Chang-C. 1986) θ d = θ d (M s, γ) (0, π 2 ) such that, when θ W (θ d, π 2 ), there exist two states (2) = (U2 a, V 2 a, pa 2, ρa 2 ) and (Ub 2, V 2 b, pb 2, ρb 2 ) such that (U2 a, V 2 a) > (Ub 2, V 2 b) and (Ub 2, V 2 b) < cb 2. Stability Criterion (C-Feldman 2005) as θ W π 2 : Choose (2) = (U2 a, V 2 a, pa 2, ρa 2 ). Detachment Criterion: There is no Regular Reflection Configuration when the wedge angle θ W (0, θ d ). Sonic Conjecture: There exists a Regular Reflection Configuration when θ W (θ s, π 2 ), for θ s > θ d such that (U2 a, V 2 a) > ca 2 at A. Incident shock Sonic Circle of (2) Reflected shock A Subsonic? _ Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 62 / 84

Detachment Criterion vs Sonic Criterion θ c > θ s : γ = 1.4 Courtesy of W. Sheng and G. Yin: ZAMP, 2008 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 63 / 84

Global Theory? (0) (1) (2) D S subsonic? Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 64 / 84

Euler Eqs. under Decomposition: (U, V ) = ϕ + W, W = 0 (ρ ϕ) + 2ρ + (ρ W ) = 0, ( 1 2 ϕ 2 + ϕ) + 1 ρ p = ( ϕ + W ) W + ( 2 ϕ + I )W, ( ϕ + W ) ω + (1 + ϕ)ω = 0 (( ϕ + W )ω) + ω = 0, ( ϕ + W ) S = 0. where S = c v ln(pρ γ ) Entropy; ω = curl W = curl(u, V ) Vorticity When ω = 0, S = const., and W = 0 on a curve transverse to the fluid direction, then, in the region of the fluid trajectories past the curve, W = 0, S = const. W = 0, p = const. ρ γ Then we obtain the Potential Flow Equation (by scaling): (ρ ϕ) + 2ρ = 0, 1 2 ( ϕ 2 + ϕ) + ργ 1 = const. > 0. γ 1 J. Hadamard: Lecons sur la Propagation des Ondes, Hermann: Paris 1903 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 65 / 84

Potential Flow Dominates the Regular Reflection, provided that ϕ C 1,1 across the Sonic Circle Potential Flow Equation { (ρ ϕ) + 2ρ = 0, 1 2 ϕ 2 + ϕ + ργ 1 γ 1 = ργ 1 0 γ 1 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 66 / 84

Potential Flow Equation (ρ( ϕ, ϕ, ρ 0 ) ϕ) + 2ρ( ϕ, ϕ, ρ 0 ) = 0 Incompressible: ρ = const. = ϕ + 2 = 0 Subsonic (Elliptic): 2 ϕ < c (ϕ, ρ 0 ) := γ + 1 (ργ 1 0 (γ 1)ϕ) Supersonic (Hyperbolic): 2 ϕ > c (ϕ, ρ 0 ) := γ + 1 (ργ 1 0 (γ 1)ϕ) Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 67 / 84

Global Theory? (0) (1) (2) D S subsonic? Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 68 / 84

Setup of the Problem for ψ := ϕ ϕ 2 in Ω div (ρ( ψ, ψ, ξ, η, ρ 0 )( ψ + v 2 (ξ, η)) + l.o.t. = 0 ψ ν wedge = 0 ψ Γsonic = 0 = ψ ν Γsonic = 0 Rankine-Hugoniot Conditions on Shock S: [ψ] S = 0 [ρ( ψ, ψ, ξ, η, ρ 0 )( ψ + v 2 (ξ, η)) ν] S = 0 ( ) ( ) Free Boundary Problem S = {ξ = f (η)} such that f C 1,α, f (0) = 0 and Ω + = {ξ > f (η)} D= {ψ < ϕ 1 ϕ 2 } D ψ satisfy the free boundary condition ( ) along S Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 69 / 84

Setup of the Problem for ψ := ϕ ϕ 2 in Ω div (ρ( ψ, ψ, ξ, η, ρ 0 )( ψ + v 2 (ξ, η)) + l.o.t. = 0 ψ ν wedge = 0 ψ Γsonic = 0 = ψ ν Γsonic = 0 Rankine-Hugoniot Conditions on Shock S: [ψ] S = 0 [ρ( ψ, ψ, ξ, η, ρ 0 )( ψ + v 2 (ξ, η)) ν] S = 0 Free Boundary Problem S = {ξ = f (η)} such that f C 1,α, f (0) = 0 and ( ) ( ) Ω + = {ξ > f (η)} D= {ψ < ϕ 1 ϕ 2 } D ψ satisfy the free boundary condition ( ) along S { solves (*) in ψ C 1,α (Ω + ) C 2 Ω+, (Ω + ) is subsonic in Ω + with (ψ, ψ ν ) Γsonic = 0, ψ ν wedge = 0 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 69 / 84

Theorem (Global Theory for Shock Reflection-Diffraction C. Feldman: Proc. Natl. Acad. Sci. USA 2005; Annals of Mathematics 2010) θ c = θ c (ρ 0, ρ 1, γ) (0, π 2 ) such that, when θ W (θ c, π 2 ), there exist (ϕ, f ) satisfying ϕ C (Ω) C 1,α ( Ω) and f C (P 1 P 2 ) C 2 ({P 1 }); ϕ C 1,1 across the sonic circle P 1 P 4 ϕ ϕ NR in W 1,1 loc as θ W π 2. Φ(t, x) = tϕ( x t ) + x 2 2t, ρ(t, x) = ( ρ γ 1 0 (γ 1)(Φ t + 1 2 Φ 2 ) ) 1 form a solution of the IBVP. γ 1 P0 Ω Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 70 / 84 ξ

Approach for the Large-Angle Case Cutoff Techniques by Shiffmanization Elliptic Free-Boundary Problem with Elliptic Degeneracy on Γ sonic Domain Decomposition Near Γ sonic Away from Γ sonic Iteration Scheme C. Feldman, J. Amer. Math. Soc. 2003 Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 71 / 84

Approach for the Large-Angle Case Cutoff Techniques by Shiffmanization Elliptic Free-Boundary Problem with Elliptic Degeneracy on Γ sonic Domain Decomposition Near Γ sonic Away from Γ sonic Iteration Scheme C. Feldman, J. Amer. Math. Soc. 2003 C 1,1 Parabolic Estimates near the Degenerate Elliptic Curve Γ sonic ; Corner Singularity Estimates In particular, when the Elliptic Degenerate Curve Γ sonic Meets the Free Boundary, i.e., the Transonic Shock Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 71 / 84

Approach for the Large-Angle Case Cutoff Techniques by Shiffmanization Elliptic Free-Boundary Problem with Elliptic Degeneracy on Γ sonic Domain Decomposition Near Γ sonic Away from Γ sonic Iteration Scheme C. Feldman, J. Amer. Math. Soc. 2003 C 1,1 Parabolic Estimates near the Degenerate Elliptic Curve Γ sonic ; Corner Singularity Estimates In particular, when the Elliptic Degenerate Curve Γ sonic Meets the Free Boundary, i.e., the Transonic Shock Removal of the Cutoff Require the Elliptic-Parabolic Estimates?? Extend the Large-Angle to the Sonic-Angle θ s?? Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 71 / 84

Elliptic Degeneracy Linear 2xψ xx + 1 c 2 2 ψ yy ψ x 0 ψ Ax 3/2 + h.o.t. when x 0 Nonlinear { (2x (γ + 1)ψx ) ψ xx + 1 c 2 2 ψ yy ψ x o(x 2 ) Ψ x=0 = 0 Ellipticity: ψ x 2x γ+1 Apriori Estimate: ψ x 4x 3(γ+1) ψ x 2 + h.o.t. when x 0 2(γ + 1) Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 72 / 84

Optimal Regularity and Sonic Conjecture Theorem (Optimal Regularity; Bae C. Feldman: Invent. Math. 2009): ϕ C 1,1 but NOT in C 2 across P 1 P 4 ; ϕ C 1,1 ({P 1 }) C 2,α ( Ω \ ({P 1 } {P 3 })) C 1,α ({P 3 }) C (Ω); f C 2 ({P 1 }) C (P 1 P 2 ). = C-Feldman 2011: The global existence and the optimal regularity hold up to the sonic wedge-angle θ s for any γ 1 for u 1 < c 1 ; u 1 c 1. (the von Neumann s sonic conjecture) P0 Ω Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 73 / 84 ξ

Existence for θ w (θ sonic, π 2 ) c 1 > u 1 Incident shock (1) (0) Sonic circle of state (2) P 1 Ω P 0 P 2 P 3 P 4 Σ (2) Sonic circle of state (1) O 1 Issues: As the wedge angle becomes smaller, prove the shock does not hit (i) Wedge boundary, (ii) Symmetry line Σ, (iii) Sonic circle B c1 (O 1 ) of state (1), where O 1 = (u 1, 0), (iv) Vertex point P 3. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 74 / 84

Existence for θ w (θ sonic, π 2 ) c 1 < u 1 Incident shock (1) (0) Sonic circle of state (2) P 1 Ω P 0 P 2 P 3 P 4 Σ Sonic circle of state (1) O 1 Issues: As the wedge angle becomes smaller, prove the shock does not hit (i) Wedge boundary, (ii) Symmetry line Σ, (iii) Sonic circle B c1 (O 1 ) of state (1), where O 1 = (u 1, 0), (iv) Vertex point P 3. This is unclear in the case c 1 < u 1. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 75 / 84

Existence for θ w (θ sonic, π 2 ) Incident shock c 1 < u 1 (1) (0) attached P 1 P 0 P 2 = P 3 P 4 Sonic circle of state (1) Is attached case possible for regular reflection? For irregular Mach reflection attached case appears to be possible, see Fig. 238 (page 144) of M. Van Dyke, An Album of Fluid Motion, The Parabolic Press: Stanford, 1982. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 76 / 84

Existence for θ w (θ sonic, π 2 ) c 1 u 1 Incident shock (1) (0) Sonic circle of state (2) P 1 Ω P 0 P 2 (2) P 4 P 3 Σ Sonic circle of state (1) O 1 Theorem (C-Feldman). If ρ 1 > ρ 0 > 0, γ > 1 satisfy u 1 c 1, then a regular reflection solution ϕ as our Theorem (2005) exists for all wedge angles θ w (θ sonic, π 2 ). The solution satisfies all properties stated in our Theorem (2005). In particular, ϕ is C 1,1 near and across the sonic arc P 1 P 4, and the shock is a C 2 curve, and ϕ 2 ϕ ϕ 1 in Ω. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 77 / 84

Existence for θ w (θ sonic, π 2 ) c 1 < u 1 Incident shock (1) (0) Sonic circle of state (2) P 1 Ω P 0 P 2 P 3 P 4 Σ Sonic circle of state (1) O 1 Theorem (C-Feldman). If ρ 1 > ρ 0 > 0, γ > 1 satisfy u 1 > c 1, then a regular reflection solution ϕ as in our Theorem (2005) exists for all wedge angles θ w (θ c, π 2 ), where -either θ c = θ sonic, -or θ c > θ sonic and for θ w = θ c there exists an attached weak solution of regular reflection-diffraction problem. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 78 / 84

Large Angle = Sonic Angle θ sonic : Admissible Solutions Incident shock S 0 S 1 Sonic circle of state (2) (1) (0) P 1 Ω P 0 P 2 P 3 P 4 Σ (2) Sonic circle of state (1) O 1 The solution ϕ is called an admissible solution if 1 ϕ C 1 (P 0 P 1 P 2 P 3 P 4 ), and P 0 P 1 P 2 is C 1 curve, 2 Equation is (strictly) elliptic in Ω \ P 1 P 4. 3 ϕ 2 ϕ ϕ 1 in Ω. 4 ϕ 1 ϕ in Ω monotonically non-increases in directions S 0 and S 1. Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 79 / 84

Large Angles = Sonic Angle θ sonic Approach: Apriori Estimates and Compactness (a) Establish the strict inequalities in (iii) and the strict monotonicities in (iv) (thus ϕ 1 ϕ strictly decreases for a cone of directions, thus the shocks are Lipschitz graphs with uniform Lip estimates) (b) Establish uniform bounds on diam(ω), ϕ C 0,1 (Ω), the monotonicities of ϕ ϕ 2 near the sonic arc; (c) Establish a uniform positive lower bound for the distance from the shock to the wedge, the sonic circle of state (1), and the uniform separation of the shock and the symmetry line; (d) Make uniform regularity estimates for the solution and its shock in weighted/scaled Hölder norms (including near the sonic arc, which imply C 1 across the sonic arc); (e) Prove that the uniform limit of admissible solutions is an admissible solution, and the uniform limit of the sequence of shocks is a shock. Continuity Method/Degree Theory= Existence of Admissible Solutions for Large Wedge Angle: = von Neumann s Sonic Conjecture Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 80 / 84

Mach Reflection: Full Euler Equations? Right space for vorticity ω?? Chord-arc z(s) = z 0 + s 0 eib(s) ds, b BMO? Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 81 / 84

Further Developments on Shock Reflection-Diffraction S.-X. Chen: Local Stability of Mach Configurations D. Serre: Multi-D Shock Interaction for a Chaplygin Gas S. Canic, B. Keyfitz, K. Jegdic, E. H. Kim: Semi-Global Solutions for Shock Reflection Problems V. Elling: Examples to the Sonic and Detachment Criteria J. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang, and Y. Zheng: Transonic Shock Formation in a Rarefaction Riemann Problem Y. Zheng+al: Pressure-Gradient Equations,?? Various Models for the Shock Reflection-Diffraction Problems?? Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 82 / 84

Multidimensional Problems vs New Mathematics Mixed and Composite Eqns. of Hyperbolic-Elliptic Type Degenerate Elliptic Techniques Degenerate Hyperbolic Techniques Transport Equations with Rough Coefficients Naturally Arising in Many Fundamental Problems in Fluid Mechanics, Differential Geometry, Optimization Elasticity, Relativity, Free Boundary Techniques Regularity Estimates when a Free Boundary Meets a Degenerate Curve Boundary Harnack Inequalities Further Understanding of Compressible Vortex Sheets and Vorticity Waves Further Analysis of Divergence-Measure Vector Fields,... New Measure-Theoretical Analysis, Geometric Measures,... More Efficient Numerical Methods,... Gui-Qiang Chen (Oxford) Multidimensional Conservation Laws September 12 16, 2011 84 / 84

Multidimensional Conservation Laws Part II: Multidimensional Models and Problems. BCAM and UPV/EHU Courses 2010/2011