An Introduction to Multidimensional Conservation Laws Part II ...

An Introduction to Multidimensional Conservation Laws Part II: Multidimensional Models and Problems Gui-Qiang Chen Department of Mathematics, Northwestern University, USA Website: http://www.math.northwestern.edu/˜gqchen/preprints

Summer Program on Nonlinear Conservation Laws and Applications Institute for Mathematics and Its Applications University of Minnesota, Minneapolis July 13–31, 2009 Gui-Qiang Chen (Northwestern)

Multidimensional Conservation Laws

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Basic References Gui-Qiang Chen Euler Equations and Related Hyperbolic Conservation Laws, In: Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 2, pp. 1-104, 2005, Eds. C. M. Dafermos and E. Feireisl, Elsevier: Amsterdam, Netherlands Gui-Qiang Chen and Mikhail Feldman Shock Reflection-Diffraction and Multidimensional Conservation Laws, In: Proceedings of the 2008 Hyperbolic Conference: Theory, Numerics, and Application, ed. E.Tadmor, AMS: Providence, 2009 Gui-Qiang Chen, Monica Torres, and William Ziemer Measure-Theoretical Analysis and Nonlinear Conservation Laws, Pure Appl. Math. Quarterly, 3 (2007), 841-879 (To Leon Simon on his 60th birthday) Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang Conservation Laws: Transonic Flow and Differential Geometry, In: Proceedings of the 2008 Hyperbolic Conference: Theory, Numerics, and Application, ed. E. Tadmor, AMS: Providence, 2009 *Download: http://www.math.northwestern.edu/˜gqchen/preprints/ Gui-Qiang Chen (Northwestern)


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∂t u + ∇ · f(u) = 0 u = (u1 , . . . , um ) ∈ Rm , x = (x1 , . . . , xd ) ∈ Rd , ∇ = (∂x1 , . . . , ∂xd ) f = (f1 , . . . , fd ) : Rm → (Rm )d is a nonlinear mapping fi : Rm → Rm for i = 1, . . . , d

∂t A(u, ut , ∇u) + ∇ · B(u, ut , ∇u) = 0 Connections and Applications: Fluid Mechanics and Related: Euler Equations and Related Equations Gas, shallow water, elastic body, combustion, MHD, .... Differential Geometry and Related: Gauss-Codazzi Equations and Related Equations Embedding, Emersion, ... Relativity and Related: Einstein Equations and Related Equations Non-vacuum states, ..... ...... Gui-Qiang Chen (Northwestern)


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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 more general problems 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, ········· Gui-Qiang Chen (Northwestern)


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Outline

1

Important Multidimensional Models

2

Multidimensional Steady Problems

3

Multidimensional Self-Similar Problems

4

Compressible Vortex Sheets and Related Problems

Gui-Qiang Chen (Northwestern)


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Unsteady Transonic Small Disturbance Equation (E-9) UTSD equation (E-9) in transonic aerodynamics: ( ∂t u + ∂x ( 12 u 2 ) + ∂y v = 0, ∂x v − ∂y u = 0, or the Zabolotskaya-Khokhlov equation (E-10): ∂x (∂t u + u∂x u) + ∂yy u = 0.



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Unsteady Transonic Small Disturbance Equation (E-9) UTSD equation (E-9) in transonic aerodynamics: ( ∂t u + ∂x ( 12 u 2 ) + ∂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).



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Unsteady Transonic Small Disturbance Equation (E-9) UTSD equation (E-9) in transonic aerodynamics: ( ∂t u + ∂x ( 12 u 2 ) + ∂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 (Northwestern)


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Steady Transonic Small Disturbance Equation (E-11) ∂x (u ∂x u) + ∂yy u = 0 or u ∂xx u + ∂yy u + (∂x u)2 = 0. Elliptic: Hyperbolic:

u>0 u 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: 2008 Gui-Qiang Chen (Northwestern)


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The Gaussian Curvature K on a Torus: Doughnut Surface or Toroidal Shell

K0 Gui-Qiang Chen (Northwestern)


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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:

¯ + ∂z f (u) = 0, ∂z = 12 (∂x − i∂y ) ∂t u

For u = u + iv and 12 f (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) = u2 = u 2 + v 2 + 2uvi, the complex Burger equation: ( ∂t u + 12 ∂x (u 2 + v 2 ) + ∂y (uv ) = 0, ∂t v − ∂x (uv ) + 12 ∂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 (Northwestern)


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Outline

1


2


3


4




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Two-D Steady Euler Equations (E-18): (x, y ) ∈ R2  ∂x (ρu) + ∂y (ρv ) = 0,    ∂ (ρu 2 + p) + ∂ (ρuv ) = 0, x y 2 + p) = 0, ∂ (ρuv ) + ∂ (ρv  x y   p ∂x ρu(E + ρ ) + ∂y ρv (E + pρ ) = 0, Constitutive Relations: 1 E = e + (u 2 + v 2 ) 2 (u, v )—fluid velocity p—scalar pressure e—internal energy θ—temperature

(e, p, θ) = (e(ρ, S), p(ρ, S), θ(ρ, S)), ρ—fluid density S—entropy For a polytropic gas, p = Rρθ,

e=

p , (γ − 1)ρ

p = p(ρ, S) = κργ e S/cv , R > 0, cv > 0, κ > 0 are constants, Gui-Qiang Chen (Northwestern)

γ =1+

R cv

κ γ−1 S/cv ρ e γ−1 γ > 1 is the adiabatic exponent

e=


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2-D Steady Full Euler Equations: Eigenvalues and Waves Eigenvalues for the 1st and 4th families: √ uv + (−1)j c u 2 + v 2 − c 2 λj = , u2 − 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 2nd and 3rd families: λi = v /u,

i = 2, 3

=⇒ Two transport equations The 2nd families: Vortex sheets The 3rd families: Entropy waves/Contact discontinuities *New Phenomena: Compressible vortex sheets Gui-Qiang Chen (Northwestern)


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Steady Potential Flow Equation (E-5)

∇ · (ρ(∇Φ)∇Φ) = 0 For a γ-law gas, p = p(ρ) = ργ /γ, γ > 1, is the normalized pressure. Then the normalized Bernoulli’s law:



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∇ · (ρ(∇Φ)∇Φ) = 0 For a γ-law gas, p = p(ρ) = ργ /γ, γ > 1, is the normalized pressure. Then the normalized Bernoulli’s law: p 1 γ − 1 2 γ−1 q ρ = ρˆ(q 2 ) := 1 − , q = u 2 + v 2 – flow speed. 2 q q 2 2 (sonic speed), q :≡ Define c = 1 − γ−1 q cr 2 γ+1 (critical speed). We rewrite Bernoulli’s law in the form 2 q 2 − qcr =


2 q2 − c 2 . γ+1


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∇ · (ρ(∇Φ)∇Φ) = 0 For a γ-law gas, p = p(ρ) = ργ /γ, γ > 1, is the normalized pressure. Then the normalized Bernoulli’s law: p 1 γ − 1 2 γ−1 q ρ = ρˆ(q 2 ) := 1 − , q = u 2 + v 2 – flow speed. 2 q q 2 2 (sonic speed), q :≡ Define c = 1 − γ−1 q cr 2 γ+1 (critical speed). We rewrite Bernoulli’s law in the form 2 q 2 − qcr =

2 q2 − c 2 . γ+1

Then the flow is subsonic (elliptic) when q < qcr , sonic (degenerate state) when q = qcr , supersonic (hyperbolic) when q > qcr . Gui-Qiang Chen (Northwestern)


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Airfoil Problems I: ∇ · (ρ(∇Φ)∇Φ) = 0, x ∈ R2 , v∞ = 0 Obstacle Boundary ∂Ω1 : Solid curve in (a); Solid closed curve in (b). Far-field Boundary ∂Ω2 : Dashed line segments in both (a) and (b). Domain Ω: bounded by ∂Ω1 and ∂Ω2 . Boundary conditions on the obstacle ∂Ω: ( ∇Φ · n = 0 on ∂Ω1 , Consistent far-field boundary conditions on ∂Ω2 , where n is the unit normal pointing into the flow region on ∂Ω. In case (b), the circulation about the boundary ∂Ω2 is zero. ∂Ω2

∂Ω2 Ω Ω

∂Ω1

∂Ω1

(a)


(b)


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Airfoil Problems II: ∇ · (ρ(∇Φ)∇Φ) = 0, x ∈ R2 , v∞ = 0

Problem 1: Existence of global entropy solutions Yes:

When the far-field velocity (u∞ , 0) with u∞ ≤ q∗ for some q∗ < qcr :

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∗ , qcr )? 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 (Northwestern)


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Wedge/Cone Problems I: Wedge Problems =⇒ Progress Wedge Problems u2

T

θc O

H

S

q

u1

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 (Northwestern)


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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



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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 (Northwestern)


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Outline

1


2


3


4




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Shock Reflection-Diffraction Problems W

W

:HGJH

? 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 (Northwestern)


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0.4108

0.4106

Subsonic

y/t

0.4104

Supersonic

0.4102

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 (Northwestern)


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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 (Northwestern)


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Scientific Issues Structure of the Shock Reflection-Diffraction Patterns Transition Criteria among the Patterns Dependence of the Patterns on the Parameters wedge angle θw , adiabatic exponent γ ≥ 1 incident-shock-wave Mach number Ms ······

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 (Northwestern)


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2-D Riemann Problem for Hyperbolic Conservation Laws ∂t U + ∇ · F (U) = 0, U3

x = (x1 , x2 ) ∈ R2 U2

t=0 U1 0 UN-1

UN

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, · · · Books and Survey Articles: Chang-Hsiao 1989, Glimm-Majda 1991 Li-Zhang-Yang 1998, Zheng 2001, Chen-Wang 2002, Serre 2005, Chen 2005,· · · Theoretical Roles: Asymptotic States and Attractors Local Structure and Building Blocks... Gui-Qiang Chen (Northwestern)


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Riemann Solutions I



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Riemann Solutions II



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Riemann Solutions vs General Entropy Solutions Asymptotic States and Attractors Observation (C–Frid 1998): Let R( xt ) be the unique piecewise Lipschitz continuous Riemann x solution with Riemann data: R|t=0 = R0 ( |x| ) Let U(t, x) be a bounded entropy solution with initial data: x U|t=0 = R0 ( )+P0 (x), R0 ∈ L∞ (S d−1 ), P0 ∈ L1 ∩L∞ (Rd ) |x| The corresponding self-similar sequence U T (t, x) := U(Tt, T x) is compact in L1loc (Rd+1 + ) Z =⇒

ess lim

t→∞ Ω

|U(t, tξ) − R(ξ)| dξ = 0

for any Ω ⊂ Rd

Building Blocks and Local Structure Local structure of entropy solutions Building blocks for numerical methods Gui-Qiang Chen (Northwestern)


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Full Euler Equations (E-1): (t, x) ∈ R3+ := (0, ∞) × R2  ∂t ρ + ∇ · (ρv) = 0    ∂t (ρv) + ∇ · (ρv ⊗ v) + ∇p = 0    ∂ ( 1 ρ|v|2 + ρe) + ∇ · ( 1 ρ|v|2 + ρe + p)v = 0 t 2 2 Constitutive Relations: p = p(ρ, e) ρ–density, v = (v1 , v2 )> –fluid velocity, p–pressure e–internal energy, θ–temperature, S–entropy For a polytropic gas:

p = (γ − 1)ρe, e = cv θ,

p = p(ρ, S) = κργ e S/cv ,

e = e(ρ, S)

γ =1+

R cv

κ γ−1 S/cv , ρ e γ−1

R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas cv > 0 is the specific heat at constant volume γ > 1 is the adiabatic exponent, κ > 0 is any constant under scaling Gui-Qiang Chen (Northwestern)


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Euler Equations for Potential Flow: (E-3): v = ∇Φ

(

∂t ρ + ∇ · (ρ∇Φ) = 0, ∂t Φ +

1 2 2 |∇Φ|

+

ργ−1 γ−1

(conservation of mass) =

ργ−1 0 γ−1 ,

(Bernoulli’s law);

or, equivalently, ∂t ρ(∂t Φ, ∇Φ, ρ0 ) + ∇ · ρ(∂t Φ, ∇Φ, ρ0 )∇Φ = 0, with

1 1 ρ(∂t Φ, ∇Φ, ρ0 ) = ργ−1 − (γ − 1)(∂t Φ + |∇Φ|2 ) γ−1 . 0 2

We will see the Euler equations for potential flow is actually EXACT in an important region of the solution to the shock reflection problem.



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Discontinuities of Solutions ∂t u + ∇x · f(u) = 0,

x ∈ Rd

An oriented surface Γ with unit normal n = (nt , n1 , · · · , nd ) ∈ Rd+1 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.

The surface (Γ, n) 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,

where (η(u), q(u)) = (−ρS, −ρvS). Gui-Qiang Chen (Northwestern)


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Shock Waves vs Vortex Sheets



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W


W

:HGJH


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Initial-Boundary Value Problem: 0 < ρ0 < ρ1 , v1 > 0 Initial condition at t = 0: ( (0, 0, p0 , ρ0 ), (v, p, ρ) = (v1 , 0, p1 , ρ1 ),

|x2 | > x1 tan θw , x1 > 0, x1 < 0;

Slip boundary condition on the wedge bdry: v · ν = 0. (1)

X2

(0)

Shock

X1

Invariant under the Self-Similar Scaling: (t, x) −→ (αt, αx), α 6= 0 Gui-Qiang Chen (Northwestern)


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Seek Self-Similar Solutions (v, p, ρ)(t, x) = (v, p, ρ)(ξ, η), (ξ, η) = ( xt1 , xt2 )   (ρU)ξ + (ρV )η + 2ρ = 0,     2   (ρU + p)ξ + (ρUV )η + 3ρU = 0,

(ρUV )ξ + (ρV 2 + p)η + 3ρV = 0,     1 γp 1 γp 1 γp   ))ξ + (V ( ρq 2 + ))η + 2( ρq 2 + ) = 0,  (U( ρq 2 + 2 γ−1 2 γ−1 2 γ−1 √ where q = U 2 + V 2 and (U, V ) = (v1 − ξ, v2 − η) is the pseudo-velocity. √ UV ±c q 2 −c 2 Eigenvalues: λ0 = VU (repeated), λ± = , U 2 −c 2 p 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 (Northwestern)


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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, p0 , ρ0 ), ξ > ξ0 , η > ξ tan θw , (U + ξ, V + η, p, ρ) → (v1 , 0, p1 , ρ1 ), ξ < ξ0 , η > 0.



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Normal Reflection When θw = 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

(2)

(1)

location of incident shock le

irc ic c

son

R elliptic sonic

circle

hyperbolic Gui-Qiang Chen (Northwestern)


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von Neumann Criteria & Conjectures (1943) Local Theory for Regular Reflection (cf. Chang-C. 1986) ∃ θd = θd (Ms , γ) ∈ (0, π2 ) such that, when θW ∈ (θd , π2 ), there exist two states (2) = (U2a , V2a , p2a , ρa2 ) and (U2b , V2b , p2b , ρb2 ) such that |(U2a , V2a )| > |(U2b , V2b )| and |(U2b , V2b )| < c2b . Stability as θW → π2 : Choose (2) = (U2a , V2a , p2a , ρa2 ).



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von Neumann Criteria & Conjectures (1943) Local Theory for Regular Reflection (cf. Chang-C. 1986) ∃ θd = θd (Ms , γ) ∈ (0, π2 ) such that, when θW ∈ (θd , π2 ), there exist two states (2) = (U2a , V2a , p2a , ρa2 ) and (U2b , V2b , p2b , ρb2 ) such that |(U2a , V2a )| > |(U2b , V2b )| and |(U2b , V2b )| < c2b . Stability as θW → π2 : Choose (2) = (U2a , V2a , p2a , ρa2 ). Detachment Conjecture: There exists a Regular Reflection Configuration when the wedge angle θW ∈ (θd , π2 ). Sonic Conjecture: There exists a Regular Reflection Configuration when θW ∈ (θs , π2 ), for θs > θd such that |(U2a , V2a )| > c2a at A.

Incident shock

A Reflected shock

Sonic Circle of (2) Subsonic?

Y_

R Gui-Qiang Chen (Northwestern)


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Detachment Criterion vs Sonic Criterion θc > θs : γ = 1.4 Courtesy of W. Sheng and G. Yin: ZAMP, 2008



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Global Theory?

(0) (1) (2)

D


S subsonic?

Y


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Euler Eqs. under Decomposition: (U, V ) = ∇ϕ + W , ∇ · W = 0  ∇ · (ρ∇ϕ) + 2ρ + ∇ · (ρ∇W ) = 0,       ∇( 1 |∇ϕ|2 + ϕ) + 1 ∇p = (∇ϕ + W ) · ∇W + (∇2 ϕ + I )W , 2 ρ    (∇ϕ + W ) · ∇ω + (1 + ∆ϕ)ω = 0 ⇐⇒ ∇ · ((∇ϕ + W )ω) + ω = 0,    (∇ϕ + W ) · ∇S = 0. S = cv 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,

where

W = 0, S = const.

⇒

W = 0, p = const. ργ

Then we obtain the Potential Flow Equation (by scaling):   ∇ · (ρ∇ϕ) + 2ρ = 0, γ−1  1 (|∇ϕ|2 + ϕ) + ρ = const. > 0. 2 γ−1

J. Hadamard: Lecons sur la Propagation des Ondes, Hermann: Paris 1903 Gui-Qiang Chen (Northwestern)


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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 2 |∇ϕ|

+ϕ+

ργ−1 γ−1

=

ρ0γ−1 γ−1

Potential Flow Gui-Qiang Chen (Northwestern)


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Potential Flow Equation

∇ · (ρ(∇ϕ, ϕ, ρ0)∇ϕ) + 2ρ(∇ϕ, ϕ, ρ0) = 0 Incompressible: ρ = const. =⇒ ∆ϕ + 2 = 0 Subsonic (Elliptic): s

2 (ργ−1 − (γ − 1)ϕ) γ+1 0

s

2 (ργ−1 − (γ − 1)ϕ) γ+1 0

|∇ϕ| < c∗ (ϕ, ρ0 ) := Supersonic (Hyperbolic): |∇ϕ| > c∗ (ϕ, ρ0 ) := Gui-Qiang Chen (Northwestern)


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Global Theory?

(0) (1) (2)

D


S subsonic?

Y


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Setup of the Problem for ψ := ϕ − ϕ2 in Ω div (ρ(∇ψ, ψ, ξ, η, ρ0 )(∇ψ + v2 − (ξ, η)) + l.o.t. = 0 ∇ψ · ν|wedge = 0 ψ|Γsonic = 0 =⇒ ψν |Γsonic = 0 Rankine-Hugoniot Conditions on Shock S: [ψ]S = 0 [ρ(∇ψ, ψ, ξ, η, ρ0 )(∇ψ + v2 − (ξ, η)) · ν]S = 0



(∗)

(∗∗)

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(∗)

(∗∗)

Free Boundary Problem ∃ S = {ξ = f (η)} such that f ∈ C 1,α , f 0 (0) = 0 and Ω+ = {ξ > f (η)} ∩ D= {ψ < ϕ1 − ϕ2 } ∩ D ψ satisfy the free boundary condition (∗∗) along S



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(∗)

(∗∗)

Free Boundary Problem ∃ S = {ξ = f (η)} such that f ∈ C 1,α , f 0 (0) = 0 and Ω+ = {ξ > f (η)} ∩ D= {ψ < ϕ1 − ϕ2 } ∩ D ψ satisfy the free boundary condition (∗∗) along S solves (*) in Ω+ , 1,α 2 ψ ∈ C (Ω+ ) ∩ C (Ω+ ) is subsonic in Ω+ with (ψ, ψν )|Γsonic = 0, Gui-Qiang Chen (Northwestern)

∇ψ · ν|wedge = 0


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Theorem (Global Theory for Shock Reflection-Diffraction C.–Feldman: PNAS 2005; Ann. Math. 2006 (accepted))

∃ θc = θc (ρ0 , ρ1 , γ) ∈ (0, π2 ) such that, when θW ∈ (θc , π2 ), there exist (ϕ, f ) satisfying ¯ and f ∈ C ∞ (P1 P2 ) ∩ C 2 ({P1 }); ϕ ∈ C ∞ (Ω) ∩ C 1,α (Ω) ϕ ∈ C 1,1 across the sonic circle P1 P4 1,1 ϕ −→ ϕNR in Wloc as θW → π2 .

1 2 γ−1 ⇒ Φ(t, x) = tϕ( xt ) + |x| − (γ − 1)(Φt + 21 |∇Φ|2 ) γ−1 2t , ρ(t, x) = ρ0 form a solution of the IBVP. ,QFLGHQW VKRFN

P0 6RQLFFLUFOH

5HIOHFWHG VKRFN

Ω ξ



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Optimal Regularity and Sonic Conjecture Theorem (Optimal Regularity; Bae–C.–Feldman: Invent. Math. 2009): ϕ ∈ C 1,1 but NOT in C 2 across P1 P4 ;

¯ \ ({P1 } ∪ {P3 })) ∩ C 1,α ({P3 }) ∩ C ∞ (Ω); ϕ ∈ C 1,1 ({P1 }) ∩ C 2,α (Ω f ∈ C 2 ({P1 })∩C ∞ (P1 P2 ).

,QFLGHQW VKRFN

P0 6RQLFFLUFOH

5HIOHFWHG VKRFN

Ω ξ



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Optimal Regularity and Sonic Conjecture Theorem (Optimal Regularity; Bae–C.–Feldman: Invent. Math. 2008): ϕ ∈ C 1,1 but NOT in C 2 across P1 P4 ;

¯ \ ({P1 } ∪ {P3 })) ∩ C 1,α ({P3 }) ∩ C ∞ (Ω); ϕ ∈ C 1,1 ({P1 }) ∩ C 2,α (Ω f ∈ C 2 ({P1 })∩C ∞ (P1 P2 ).

=⇒ The optimal regularity and the global existence hold up to the sonic wedge-angle θs for any γ ≥ 1 for u1 < c1 ; u1 ≥ c1 . (the von Neumann’s sonic conjecture) ,QFLGHQW VKRFN

P0 6RQLFFLUFOH

5HIOHFWHG VKRFN

Ω ξ



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Approach for the Large-Angle Case Cutoff Techniques by Shiffmanization ⇒ Elliptic Free-Boundary Problem with Elliptic Degeneracy on Γsonic

Domain Decomposition Near Γsonic Away from Γsonic

Iteration Scheme C.–Feldman, J. Amer. Math. Soc. 2003



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Iteration Scheme C.–Feldman, J. Amer. Math. Soc. 2003 1,1

C Parabolic Estimates near the Degenerate Elliptic Curve Γsonic ; Corner Singularity Estimates In particular, when the Elliptic Degenerate Curve Γsonic Meets the Free Boundary, i.e., the Transonic Shock



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Iteration Scheme C.–Feldman, J. Amer. Math. Soc. 2003 1,1

C Parabolic Estimates near the Degenerate Elliptic Curve Γsonic ; Corner Singularity Estimates In particular, when the Elliptic Degenerate Curve Γsonic 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 θs ??



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Elliptic Degeneracy Linear 2xψxx +

1 ψyy − ψx ∼ 0 c22

ψ ∼ Ax 3/2 + h.o.t. Nonlinear ( (2x − (γ + 1)ψx ) ψxx + Ψ|x=0 = 0 Ellipticity: Apriori Estimate:

when x ∼ 0 1 ψ −ψx c22 yy

2x ψx ≤ γ+1 4x |ψx | ≤ 3(γ+1)

x2 ψ∼ + h.o.t. 2(γ + 1) Gui-Qiang Chen (Northwestern)

∼ o(x 2 )


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Large Angles =⇒ Sonic Angle θs : Method of Continuity It suffices to prove that, for any θc ∈ (θs , π2 ), a global regular reflection solution exists for all wedge angles θw ∈ [θc , π2 ). Design M ⊂ [θc , π2 ) to be a set of all wedge angles for which a global regular reflection solution exists and satisfies certain properties; Show that the set M is nonempty ⇐= Existence result for the large-angle case;



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Large Angles =⇒ Sonic Angle θs : Method of Continuity It suffices to prove that, for any θc ∈ (θs , π2 ), a global regular reflection solution exists for all wedge angles θw ∈ [θc , π2 ). Design M ⊂ [θc , π2 ) to be a set of all wedge angles for which a global regular reflection solution exists and satisfies certain properties; Show that the set M is nonempty ⇐= Existence result for the large-angle case; Prove that M is relatively open in [θc , π2 ); Prove that M is relatively closed in [θc , π2 ); =⇒ M = [θc , π2 ). =⇒ von Neumann’s Sonic Conjecture Gui-Qiang Chen (Northwestern)


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Other Related Works on the Potential Flow Equations Majda-Thomann: Stability of multidimensional shock waves, CPDE, 1987 Glimm-Majda: IMA Volume, 1991 Proceedings of the IMA Workshops Morawetz: CPAM 1994 Shock reflection patterns via asymptotic analysis Elling-Liu: Elliptic principle, JHDE 2006 Supersonic flow onto a solid wedge, CPAM 2008 (Prandtl conjecture)

Recent Research Activities · · · · · · =⇒ Mikhail Feldman’s Lecture on Monday, July 20 Gui-Qiang Chen (Northwestern)


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Mach Reflection: Full Euler Equations

? Right space for vorticity ω? Rs ? Chord-arc z(s) = z0 + 0 e ib(s) ds, b ∈ BMO? Gui-Qiang Chen (Northwestern)


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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??



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Shock Reflection-Diffraction vs New Mathematics Free Boundary Techniques Mixed and Composite Eqns. of Hyperbolic-Elliptic Type Degenerate Elliptic Techniques Degenerate Hyperbolic Techniques Transport Equations with Rough Coefficients

Regularity Estimates when a Free Boundary Meets a Degenerate Curve Boundary Harnack Inequalities Spaces for Compressible Vortex Sheets More Efficient Numerical Methods · · · ······



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Two-Dimensional Riemann Problems ∂t U + ∇ · F (U) = 0, U3

x = (x1 , x2 ) ∈ R2 U2

t=0 U1 0 UN-1

UN

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, · · · Books and Survey Articles: Chang-Hsiao 1989, Glimm-Majda 1991 Li-Zhang-Yang 1998, Zheng 2001, Chen-Wang 2002, Serre 2005, Chen 2005,· · · Theoretical Roles: Asymptotic States and Attractors Local Structure and Building Blocks... Gui-Qiang Chen (Northwestern)


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Outline

1


2


3


4




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Compressible Vortex Sheets/Entropy Waves

Lecture 3



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