Linear and Nonlinear Waves in Gas Dynamics Barbara Lee Keyfitz The Ohio State University bkeyfitz@math.ohio-state.edu April 2, 2016 Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 1 / 28
Outline 1 Background Characteristics 2 The Problem Explanation and People 3 Wave Equations Basic Linear Quasilinear: Gas Dynamics 4 Compressible Gas Dynamics System Self-Similar Solutions Free Boundary Problem and Local Solution Mysteries and Constraints 5 Recent Results Incompressible Flow: Himonas & Misio� lek Compressible Flow Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 2 / 28
Background Characteristics An Example PDE ∂ u ∂ t + a ∂ u ∂ x = 0 General solution ( f ∈ C 1 classical solution; f ∈ D ′ weak solution) u ( x , t ) = f ( x − at ) Characteristic curves x − at = const are significant: propagation of signals separation of regions of smooth flow unsuitable for prescribing data We see geometry in R 2 = ( x , t ): tangents to char are � t = ( a , 1) geometry in dual space R 2 : char normals � ν = (1 , − a ) = ( ξ, τ ) satisfy τ + a ξ = 0; recall principal symbol of ∂ t + a ∂ x is τ + a ξ Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 3 / 28
Background Characteristics What You Saw Was an Example of A first-order system: d � ∂ u A j + b = 0 ∂ x j 0 A j are n × n matrices, u , b are n -vectors L 0 = � d 0 A j ξ j is the principal symbol (matrix), ν = ( ξ 0 , . . . , ξ d ) to det ( � A j ξ j ) = 0 is characteristic normal soln � surfaces in R d +1 = { ( x 0 , . . . , x d ) } whose normals are characteristic are characteristic surfaces we will have different stories for A j constant, A j = A j ( x ) (linear system) and A j = A j ( x , u ) (quasilinear system) Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 4 / 28
Background Characteristics Significance Cauchy-Kovalevsky Theorem: locally, and for analytic functions, data on a non-characteristic surface ⇒ ∃ ! analytic solution Classification of PDE via characteristics: elliptic (no real characteristics) hyperbolic (maximal set of real characteristics) ‘parabolic’: borderline, in between, mixed, dispersive etc Hyperbolic system (with distinguished time variable) d � ∂ u ∂ u A 0 ∂ t + A j + b = 0 ∂ x j 1 A j = A j ( x , t , u ), b = b ( x , t , u ) (quasilinear) det ( τ I + � A − 1 0 A j ξ j ) = 0 ⇒ τ i ( ξ ) real (characteristic normals) Symmetrizable hyperbolic: A 0 symm, pos def; A j symm, ∀ j Cauchy problem for linear hyperbolic systems well-posed in H s Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 5 / 28
The Problem Explanation and People The Focus of this Talk: Quasilinear Systems We are faced with some incompatible difficulties Local (=short time) existence for quasilinear (QL) symmetrizable hyperbolic systems in H s , s > d / 2 + 1 Life-span of H s solutions depends on � u 0 � s (and on A j ) Solutions of QL systems don’t stay in H s (e.g., Burgers Equation): 1 u t + uu x = 0 , u ( x , 0) = u 0 , u ( x + u 0 ( x ) t , t ) = u 0 ( x ) , t < − u ′ 0 ( x ) If d = 1 and � u 0 � BV < δ 0 , ∃ ! global (in t ) solution in BV (= W 1 , ∞ ) for QL systems ∂ t u + ∂ x f ( u ) = 0 with A ( u ) = df strictly hyperbolic + For d ≥ 2, only theory is short-time result in H s ; no other theory Theorem (Littman, Brenner, Rauch): Do not expect well-posedness in W m , p for p � = 2 Our idea: explore what goes wrong for some examples Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 6 / 28
The Problem Explanation and People People Working in this Direction My co-authors Other people and groups (incomplete list) Suncica Canic Gui Qiang Chen John Holmes Mikhail Feldman Gary Lieberman Tai Ping Liu David Wagner Volker Elling Eun Heui Kim Yuxi Zheng Allen Tesdall Marshall Slemrod Katarina Jegdic Dehua Wang Hao Ying Feride Tiglay Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 7 / 28
The Problem Explanation and People Target Equations Equations of Ideal Compressible Gas Dynamics in Two Space Dimensions System of 4 equations Important and well-studied (engineering and computation) Comparison possible with simplified models Initial approach looked at selfsimilar solutions ( x t and y t ) Leads to tractable problems (and insight into what might go wrong) Self-similar approach ⇒ interesting analysis Also look beyond self-similar problems Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 8 / 28
Wave Equations Basic Linear Warm-up Problem: Linear Wave Equation Example: Wave eqn in 2-D: u tt = c 2 ( u xx + u yy ) As a system u 1 = u y , u 2 = u t − cu x � u 1 � � 1 � � u 1 � � 0 � � u 1 � 0 1 ∂ t = c ∂ x + c ∂ y u 2 0 − 1 u 2 1 0 u 2 � � � η ξ 2 + η 2 ); � Normals � ν = ( ξ, η, ± c Char vbles v = ξ 2 + η 2 ξ ∓ ν ) ∈ R n | L 0 v = 0 } Characteristic variables: eigenvectors { v = v ( � Characteristic normals: τ 2 = c 2 ( ξ 2 + η 2 ), the “characteristic cone” � ξ 2 + η 2 t = 0 Characteristic surfaces: planes ξ x + η y ± c Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 9 / 28
Wave Equations Basic Linear Fundamental Solution to Linear Wave Equation Envelope of characteristic surfaces through (0 , 0 , 0) is the “wave cone”: x 2 + y 2 = c 2 t 2 Boundary of domain of influence of the origin Support cone of fundamental solution (support includes interior) Singular support is boundary of cone Data u ( x , y , 0) = 0, u t ( x , y , 0) = u 0 ( x , y ): � 1 u 0 ( ξ, η ) u ( t , x , y ) = � c 2 t 2 − ( x − ξ ) 2 − ( y − η ) 2 d ξ d η = K W ∗ u 0 4 π c B where B = { ( ξ, η ) | ( x − ξ ) 2 + ( y − η ) 2 ≤ c 2 t 2 } K W ( · , t ) ∈ H − d / 2+1 − ε Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 10 / 28
Wave Equations Quasilinear: Gas Dynamics Characteristic Structure of Gas Dynamics Equations Full Euler Equations: Adiabatic, compressible, ideal gas dynamics “conserved quantities” “fluxes” ρ t + ( ρ u ) x + ( ρ v ) y = 0 ( ρ u ) t + ( ρ u 2 + p ) x + ( ρ uv ) y = 0 ( ρ v ) t + ( ρ uv ) x + ( ρ v 2 + p ) y = 0 ( ρ E ) t + ( ρ uH ) x + ( ρ vH ) y = 0 State variables ρ (density), ( u , v ) (velocity), and p (pressure) 1 p ρ + 1 γ p ρ + 1 2( u 2 + v 2 ) , 2( u 2 + v 2 ) E = H = γ − 1 γ − 1 γ ≈ 1 . 4: parameter (ratio of specific heats) Linearize at a state u = ( ρ, u , v , p ) Roots of characteristic equation (det L 0 = 0) – real; a double root: � � ξ 2 + η 2 , where ¯ ¯ τ = 0 , 0 , ± γ p /ρ τ = τ − ( ξ u + η v ) Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 11 / 28
Wave Equations Quasilinear: Gas Dynamics Characteristic Normals, Familiar and Unfamiliar Four families fall into two types; fix u = ( ρ, u , v , p ) to study. (1) τ = ( ξ u + η v ): all normals lie in a plane with normal ( u , v , − 1). Corresponding char surfaces all contain ( u , v , − 1) (their envelope) Dynamic behavior is “transport” equation Domain of influence of (0 , 0 , 0) is the line ( − ut , − vt , t ). � � ξ 2 + η 2 (2) pair τ = ( ξ u + η v ) ± γ p /ρ Normals form pair of conical surfaces (forward and backward) like WE Wave cone, envelope of corresponding char surfaces, is tilted cone (tilt depends on u , v ) with size (“speed”) depending on p and ρ ) Note (1) is also a double characteristic Contrast (1) and (2) in two ways: “transport” vs “wave equation” and “nonlinear” (propagation speed depends on states) vs “linearly degenerate” ( ∇ τ ( u , ξ ) · r ( u , ξ ) = 0) Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 12 / 28
Wave Equations Quasilinear: Gas Dynamics Summary of the Structure Pair: t � � ξ 2 + η 2 τ = ( ξ u + η v ) ± γ p /ρ – acoustic waves – “wave-equation-like” y x propagation – genuinely nonlinear (Burgers) Pair: τ = ξ u + η v (double) – entropy and vorticity waves – transport-equation-like Characteristic Normals propagation – linearly degenerate For general QL systems, other possibilities exist, but they may not correspond to anything that occurs in physical models Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 13 / 28
Compressible Gas Dynamics System Self-Similar Solutions Self-similar Problems: 2-D Riemann Problems (with ˇ Cani´ c, Lieberman, Kim, Jegdic, Tesdall, Popivanov, Payne, Ying) • Analogy with 1-D: focus on transport and wave interactions • Benchmark problem: Shock reflection by a wedge Incident Shock Flow S= t Σ Wedge Reflected X= t Ξ Shock t<0 t=0 t>0 • Work in self-similar coordinates: ξ = x t , η = y t • Reduced eq’n ( − ξ + A ( U )) U ξ + ( − η + B ( U )) U η = 0 • Type Changes: hyperbolic for ( ξ, η ) >> 1; ‘subsonic’ region near 0 • Change of Type in Nondegenerate Waves Only Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 14 / 28
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