Using Geometric Singular Perturbation Theory to Understand Singular - PowerPoint PPT Presentation

Using Geometric Singular Perturbation Theory to Understand Singular Shocks Barbara Lee Keyfitz The Ohio State University bkeyfitz@math.ohio-state.edu June 26, 2012 HYPE 2012 Padova Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 1 / 17

Outline 1 Conservation Laws and Their Pathologies 2 The Problem We Would Like to Solve: Two-Component Chromatography 3 The Problem We Did Solve: Gas Dynamics, Conserving the Wrong Variables Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 2 / 17

Background Conservation Laws and Their Pathologies Our focus, in U t + F ( U ) x ≡ U t + A ( U ) U x = 0, λ i ( A ( U )) real Dependence of characteristic speeds on state U Example: Burgers equation, u t + uu x = 0, λ = u Systems exhibit more complicated dependence(s) than do scalar equations Weak solutions are standard � Weak form of the system U φ t + F ( U ) φ x = 0 Bounded, piecewise smooth solutions exhibit shocks that satisfy Rankine-Hugoniot relation s [ U ] = [ F ( U )] Low-regularity solutions: singular shocks Are not locally bounded Do not satisfy RH relation Satisfy the equation in an even weaker sense (theory by Sever) Some examples can be described by distributions Are best understood by means of approximations Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 3 / 17

Chromatography Conservation Law Models for Chromatography Two components (concentration u i for chemical i ); total mass conserved ∂ + ∂ � � u i + v i ( u ) ∂ x u i = 0 , i = 1 , 2 ∂ t Forced at constant velocity through a column packed with a solid (‘fixed bed’) onto which they are adsorbed Neglect: heat cond., diffusion, viscosity & finite rate of adsorption System in thermal and chemical equilibrium Amount of chemical i adsorbed is v i ( u 1 , u 2 ) v i obtained from adsorption laws (linear rates) dv dt = k 1 c ( V − v ) − k 2 v At equilibrium, dv / dt = 0, non-dimensionalized functions are a i u i v i = 1 + u 1 + u 2 Langmuir kinetics: Components compete at different rates, a 1 < a 2 Classical and well-studied system Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 4 / 17

Chromatography ‘Generalized Langmuir’ kinetics of Marco Mazzotti, ETH u 2 New model for v : a i u i v i = 1 − u 1 + u 2 NH replaces a i u i v i = 1 + u 1 + u 2 u Physically represents ‘cooperation’ 1 rather than competition for sites Findings System not hyperbolic for some (physically realizable) states Restrict to hyperbolic region near 0 Not all Riemann problems have solutions Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 5 / 17

Chromatography What Happens? Simulation (phase plane) by Mazzotti Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 6 / 17

Chromatography Appearance of Singular Shocks Simulation (Mazzotti) Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 7 / 17

Chromatography Experimental Appearance of Singular Shocks Experiments (Mazzotti et al) Components phenetole (C 8 H 10 O) and 4-tert-butylphenol (C 10 H 14 O) Selected to give (1) cooperation in adsorption rather than competition and (2) linear adsorption rates at experimental concentrations Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 8 / 17

A Solved Problem The Velocity-Entropy System of Isentropic Gas Dynamics Joint work with Charis Tsikkou, to appear in QAM (following Schecter) � ρ t + ( u 1 ρ ) x = 0 U A ( ρ u 1 ) t + ( ρ u 2 1 + A ρ γ ) x = 0 , q ( ρ ) = A γ ρ γ − 1 U B γ − 1 = ρ γ − 1 u 2 = 2 − γ S 1 u 2 1 − q 2 S 2 B S 2 1 < γ < 5 / 3 R 2 (U L ) R 1 (U L ) S 1 U L  u 1 t + ( (3 − γ ) u 2 1 − u 2 ) x = 0  2  u 2 t + [ (2 − γ )(5 − 3 γ ) u 3  1 + ( γ − 1) u 1 u 2 ] x = 0 .  6 Nonhyperbolic region (above B ) Compact Hugoniot locus Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 9 / 17

A Solved Problem Region of Classical Riemann Solutions 5 ↓ Region 1: 1-shock ⇒ 2-shock U A R 1 (U L ) Region 2: 1-rarefaction ⇒ 2-rarefaction 3 B 2 S 2 Region 3: 1-rarefaction ⇒ 2-shock U D ↓ R 2 (U L ) J 1 1 U C S 1 U L Region 4: 1-shock ⇒ 2-rarefaction 4 6 Region 5: 1-rarefaction ⇒ vacuum R 2 (U C ) state ⇒ 2-rarefaction Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 10 / 17

A Solved Problem Approximation by Dafermos Regularization ε tU xx = U t + F ( U ) x 2 ξ = x y 2 =c + y 1 y 2 t 2 y 2 =c − y 1 ε d 2 U � � dU d ξ 2 = DF ( U ) − ξ I d ξ BC U ( −∞ ) = U L , U (+ ∞ ) = U R ε p y 1 ( ξ − s 1   ε q ) U ( ξ ) =   ε r y 2 ( ξ − s 1 y 1 ε q ) η = ξ − s ε q ; p = 1, q = 2 = r ; dY d η = F ( Y ) Inner part/Outer part Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 11 / 17

A Solved Problem Existence of Profiles via Geometric Singular Perturbation Theory: Krupa, Szmolyan & Schecter GSPT answers questions: How is the singular part of the solution (the homoclinic orbit) connected to the outer part (constant states)? What happens to the RH relation? What is limiting process ε → 0? then W ′ = − U and system is U t + F ( U ) x = ε tU xx ε U ′ = F ( U ) − ξ U − W Self-similar ξ = x W ′ = − U t ξ ′ = 1 ε U ′′ = − ξ U ′ + F ( U ) ′ = ( − ξ U ) ′ + U + F ( U ) ′ Example of a fast-slow system ε x ′ = f ( x , y , ε ) Define y ′ = g ( x , y , ε ) W ≡ F ( U ) − ξ U − ε U ′ Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 12 / 17

A Solved Problem The Idea Fast time τ = θ/ε Our system uses τ : � ε x ′ = f ( x , y , ε ) System ˙ U = F ( U ) − ξ U − W y ′ = g ( x , y , ε ) ˙ 1 Solve in slow time W = − ε U � 0 = f ( x , y , 0) ˙ ε = 0 ξ = ε y ′ = g ( x , y , 0) 2 Solve in fast time � ˙ with invariant sets = f ( x , y , 0) x ε = 0 y ˙ = 0 { W = F ( U ) − ξ U } 3 Show that these singular orbits and scaling with η = τ/ε and are connected if ε > 0 Y = diag { ε, ε 2 } U : 4 To use ‘Fenichel Theory’, which requires normally hyperbolic Y ′ = F ( Y ) invariant manifolds (orbits), W ′ = − diag { 0 , 1 } Y ‘blow up’ some orbits if ξ ′ = 0 necessary Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 13 / 17

A Solved Problem Reduced System Blow up of eq’m E = { Y = 0 , ε = 0 } Coord chart on blown up surface: r 2 ) Y , ε = ¯ Y = diag(¯ r , ¯ r ¯ ε | Y | 2 + ¯ ε 2 = 1  y 1 ¯ y 1 u 1 a = y 2 = √ y 2 = √ ¯ √ u 2   r 2 = ¯ r 2 ¯ y 2 = y 2 = ε 2 u 2 2 y 2 =c + y 1 y 2 2 y 2 =c − y 1 ¯ ε ε 1 b = y 2 = √ y 2 =  √ ¯ √ u 2  a ′ =(2 − γ ) a 2 − 1 − (2 − γ )(5 − 3 γ ) a 4 12 y 1 + b � − ξ a − 2 bw 1 + b 2 aw 2 � 2 � (2 − γ )(5 − 3 γ ) r ′ = r a 3 − 3 b ξ Outer system U → ∞ in finite time 6 2 Inner system Y ⇒ homoclinic orbit +3( γ − 1) a − 3 b 3 w 2 � � 0 ξ ′ = rb 2 w ′ w ′ 1 = − rab , 2 = − r , [ W ] = � (2 − γ )(5 − 3 γ ) b ′ = − b a 3 − 3 b ξ � y 2 6 2 +3( γ − 1) a − 3 b 3 w 2 � so one RH cond holds, not both Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 14 / 17

A Solved Problem Overview of the Singular Trajectories ε ¯ b W s ( T 0 2 ( U R )) , ξ > s singular a 3 & a 2 : equilibria of W s ( N 0 2 ( U R )) , ξ > s singular U R { a , b , r } system W u ( T 0 0 ( U L )) , ξ < s singular Verify W u ( N 0 0 ( U L )) , ξ < s singular W u ( q R ) normal U L ξ W s ( q L ) hyperbolicity a 1 a 2 , ξ = s singular , q R a a 4 transversality a 3 W s ( C 2 ) ξ = s singular W u ( C 3 ) hypotheses of q L r corner lemma y 2 role of strict S 2 u overcompressibil- λ 2 ( U ) < ξ ity S 0 ξ < λ 1 ( U ) Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 15 / 17

A Solved Problem The Result Theorem In the velocity-entropy system with 1 < γ < 5 / 3 , assume that U R is in the interior of region 7 with respect to U L , so that with s singular ( U L , U R ) ≡ F 1 ( U L ) − F 2 ( U R ) , u L 1 − u R 1 we have 0 < F 2 ( U L ) − F 2 ( U R ) − s singular ( u L 2 − u R 2 ) , and the strict inequalities 1 s singular ( U L , U R ) < λ 1 ( U L ) 2 λ 2 ( U R ) < s singular ( U L , U R ) hold. Then there exists a singular shock connecting U L and U R ; that is, a solution U ε of the Dafermos regularization which becomes unbounded as ε → 0 . Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 16 / 17

A Solved Problem Summary Original model problem (isothermal gas dynamics, γ = 1) led to discovery of weak solutions of very low regularity (measures) Theory developed by Sever for systems with this structure Sever’s theory is based on distributions ( δ -functions) GSPT, developed by Fenichel, Kopell, Kaper, Jones, Krupa, Szmolyan, and others, provides insight into structure of singular solutions (more detail than distributions) Other approaches given by generalized distribution theory of Colombeau et al Recent model from chromatography has physical significance, and cannot be analysed via classical distributions Analysis of chromatography system using GSPT is in progress (with Ting-Hao Hsu, Martin Krupa, and Charis Tsikkou) Barbara Keyfitz (Ohio State) GSPT and Singular Shocks HYPE 2012 Padova 17 / 17