C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Wave motion around 2-D viscous shock profile Shih-Hsien Yu Department of Mathematics, National University of Singapore 24th Annual Workshop on Differential Equations 24th, National Sun Yat-sen University, Kaoshiong January 22-23, 2016
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Viscous shock layer stability An example: • A compressible Navier-Stokes equation in 2-D, � x ∈ R 2 . � � ρ t + ∇ · � u = 0 , ρ ( � x , t ) : density at ( � x , t ) , x , t ) ∈ R 2 : fluid velocity at ( � m t + ∇ · � � u ⊗ � m + ∇ p ( ρ ) − ∆ � � u ( � x , t ) u = 0 � � m ≡ ρ� u : momentum . p ( ρ ) = ρ γ : pressure , γ ∈ [ 1 , 5 / 3 ) • A viscous shock layer Ψ( x − st ) : A travelling wave soluton. � ρ ± � � ( ρ − , ρ + ) , ( � m − , � m + ) : end states of a shock wave lim x →±∞ Ψ( x ) = , � m ± s : Speed of the shock wave • Time-asymptotic stability of viscous shock layer To study ( ρ, � m ) t ( � x , t ) − Ψ( x − st ) as t → ∞ .
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Physics and Solution of PDE Relationship in Classical PDE-Physics • Physics = ⇒ PDE (model of physics) • Solution of PDE = ⇒ Realization of Physics. Practically no relationship in Modern PDE-Physics Introduction of Real analysis = ⇒ Solutions of PDE(PDE not necessary physical) � = ⇒ Realization of Physics Viscous shock layer stablity in 2-D : A platform for exploring new tools for studying PDE
Sound Wave C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Illustrations A Simplifed Hyperbolic Diagram Fluid velocity Fluid velocity Sound wave refmection Sound wave refmection Supersonic Region Subsonic Region Shock wave front
Sound Wave C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Illustrations Realistic Diagram Fluid velocity Fluid velocity Sound wave refmection Sound wave refmection Supersonic Region Subsonic Region Shock wave front
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Matheimatics-Physics natures in viscous conservation laws • When space-dimension=1, modern PDE is sufficient to show nonlinear time-asymptotic stability of a viscous shock profile. • When space-dimension ≥ 2 , perturbations in subsonic region progress after the light-cone in space-time domain within a light-cone , and the perturbations with a light-cone decay with a slower algebraic rate . This slowly decaying property causes modern PDE not sufficient for studying nonlinear coupling and nonlinear time-asymptotic stablity. ” of classical PDE shall remain and evolve with development of • “ Modern PDE . To construct the solutions of PDE with further physics natures of the solutions such as the pointwise in the space-time domain, etc.. The key ingredient should be a better realization in physics domain and in transform variables of “Green’s function”
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Model nonlinear problem in 2-D � U t + UU x + V y = ∆ U V t + V x + U y = ∆ V � � U � � H ( x ) � 1 for x < 0 • shock wave = , H ( x ) ≡ V 0 − for x > 0 � U � � ϕ ( x ) � • = , ϕ ( x ) ≡ − tanh ( x / 2 ) shock profile V 0
Sound Wave C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D An illustration A simplifed diagram for basic elements Fluid velocity Fluid velocity Sound wave refmection Sound wave refmection Supersonic Region Subsonic Region 2 U 0 2 U 2 t 0 t x y Shock wave front
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Five basic problems PLUS auxiliary problem I . For two far fields � � �� � � �� 0 ∂ y − ∂ x ∂ y ∂ t + ∂ x − ∆ + U − ( x , y , t ) = 0 ∂ t − ∆ + U + ( x , y , t ) = 0 ∂ y 0 ∂ y ∂ x � � � � 1 0 1 0 U − ( x , y , 0 ) = δ ( x ) δ ( y ) U + ( x , y , 0 ) = δ ( x ) δ ( y ) 0 1 0 1 II . For interactions between two ends of a shock wave � � �� � � �� ∂ x H ( x ) ∂ y H ( x ) ∂ x ∂ y ∂ t − ∆ + G ( x , y , t ; x ∗ ) = 0 ∂ t − ∆ + G ( x , y , t ; x ∗ ) = 0 ∂ y ∂ x ∂ y ∂ x � � � � 1 0 1 0 G ( x , y , 0 ; x ∗ ) = δ ( x − x ∗ ) δ ( y ) G ( x , y , 0 ; x ∗ ) = δ ( x − x ∗ ) δ ( y ) 0 1 0 1 III . For interactions between shock profile and a shock wave e − ( x − x ∗ ) 2 + y 2 � − t ( ∂ t − ∆ + ϕ ( x ) ∂ x ) g ( x , y , t ; x ∗ ) = 0 g ( x , y , t ; x ∗ ) = cosh ( x ∗ / 2 ) 4 t 4 cosh ( x / 2 ) 4 π t g ( x , y , 0 ; x ∗ ) = δ ( x − x ∗ ) δ ( y ) IV . Auxilary problem � � �� ϕ ( x ) ∂ x ∂ y ∂ t − ∆ + G ( x , y , t ; x ∗ ) = 0 ∂ y ∂ x � � 1 0 G ( x , y , 0 ; x ∗ ) = δ ( x − x ∗ ) δ ( y ) 0 1
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Interactions at shock front • ∂ t and ∂ y are tangential to shock front • ∂ t and ∂ y can be replaced by transform variables s and η • ∂ x is normal to shock front and local � ∞ � e − i η y − st f ( x , y , t ) dtdy . • Laplace-Fourier transform: L [ f ]( x , i η, s ) ≡ R 0 � � �� � � 0 i η 1 0 s + η 2 + ∂ x − ∂ 2 x + L [ U − ] = δ ( x ) i η 0 0 1 I . � � �� � � − ∂ x i η 1 0 s + η 2 − ∂ 2 x + L [ U + ] = δ ( x ) i η ∂ x 0 1 � � �� � � ∂ x H ( x ) i η 1 0 s + η 2 − ∂ 2 x + L [ G ] = δ ( x − x ∗ ) i η ∂ x 0 1 II . � � �� � � H ( x ) ∂ x i η 1 0 s + η 2 − ∂ 2 x + L [ G ] = δ ( x − x ∗ ) i η ∂ x 0 1 Interaction by II ⇒ � � �� H ( x ) 0 Both L [ G ] and − ∂ x + L [ G ] are continuous in x . 0 1 Both L [ G ] and ∂ x L [ G ] are continuous in x .
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Interaction ⇒ Scattering data � � ∂ x H ( x ) �� � 1 � i η 0 s + η 2 − ∂ 2 x + L [ G ] = δ ( x − x ∗ ) i η ∂ x 0 1 2 2 � � T + kl ( i η, s ) e − λ + , k x ∗ + λ − , l x R + kl ( i η, s ) e − λ + , k x ∗ − λ + , l x L [ G ]( x , i η, s ; x ∗ ) = k , l = 1 k , l = 1 L [ U + ]( x − x ∗ , i η, s ) x=0 x >0 R ± kl : Reflective matrix * T ± : Transmissive matrix kl 2 2 � � R − kl ( i η, s ) e λ − , k x ∗ + λ − , l x T − kl ( i η, s ) e λ − , k x ∗ − λ + , l x L [ G ]( x , i η, s ; x ∗ ) = k , l = 1 k , l = 1 L [ U − ]( x − x ∗ , i η, s ) x=0 x <0 *
C ONSERVATION LAWS I NTERACTION DYNAMICS IN MULTI -D Laplace wave numbers • Characteristic Polynomials for ODE. � s + η 2 − ξ 2 ∓ ξ � i η P ± ( ξ ; i η, s ) = det s + η 2 − ξ 2 + ξ i η • Eight basic “Laplace” wave numbers λ ± , j ( i η, s ) : P ± ( λ ± , j ; i η, s ) = 0 . ( λ ± , j is a complex wave number and P ± ( λ ; i η, s ) = 0 is an implicit dispersive relationship between the Laplace wave number and s complex frequency. ) { λ + , 1 , λ + , 2 , − λ + , 1 , − λ + , 2 } , { λ − , 1 , λ − , 2 , λ − , 3 , λ − , 4 } = 1 / 2 + { σ + , σ − , − σ + , − σ − } , � � ( s + 1 / 4 ) − 1 / 2 ) 2 + η 2 , λ + , 1 = ( � � ( s + 1 / 4 ) + 1 / 2 ) 2 + η 2 , λ + , 2 = ( � ( 1 / 4 + s + η 2 ± i η ) . σ ± = � • Singular wave number: Λ ≡ λ + , 1 − s + 1 / 4 + 1 / 2 . • Four “Laplace” wave trains. ( In contrast to wave train e − i κ x + i ω ( κ ) t ) � e − λ + , 1 ( i η, s ) x , e − λ + , 2 ( i η, s ) x for x > 0 , e λ − , 1 ( i η, s ) x , e λ − , 2 ( i η, s ) x for x < 0 .
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