Shock Waves for Conservation Laws With Physical Viscosity Tai-Ping Liu Academia Sinica, Taiwan Stanford University In Honor of 60th Birthday of Alberto Bressan June 13-June 17, 2016, SISSA Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Conservation laws with physical viscosity u t + f ( u ) x = ( B ( u ) u x ) x , u ∈ R n . Examples: p-system, n = 2, v t − u x = 0 u t + p ( v ) x = ( ε ( v ) u x ) x . Compressible Navier-Stokes equations, n = 3, ρ t + ( ρ u ) x = 0 , continuity equation , ( ρ u ) t + ( ρ u 2 + p ) x = ( µ u x ) x , momentum equations . ( ρ E ) t + ( ρ Eu + pu ) x = ( µ uu x + κθ x ) x , energy equation . MHD, Visco-elasticy, n = 7. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Conservation laws with physical viscosity u t + f ( u ) x = ( B ( u ) u x ) x . Kawashima-Shizuta Systems: Assumption 1 : The system has a strictly convex entropy η .That is, there exists an entropy pair ( η ( u ) , F ( u )) , such that η ( u ) is strictly convex, M 0 ( u ) ≡ ∇ 2 η ( u ) > 0 , ( ∇ η ) f ′ = ∇ F , and ( ∇ 2 η ) B ≥ 0 . The above physical systems are endowed with convex entropy. This is needed for the local existence theory. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity u t + f ( u ) x = ( B ( u ) u x ) x . Kawashima-Shizuta Systems: Assumption 2 : The viscosity matrix B ( u ) is nonzero and there exists a smooth one-to-one mapping u = g (˜ u ) such that the null space K of ˜ u )) g ′ (˜ B (˜ u ) ≡ B ( g (˜ u ) u . Also, K ⊥ is invariant under is independent of ˜ u ) maps R n to K ⊥ . u )) , and ˜ g ′ (˜ u ) t M 0 ( g (˜ B (˜ That is, by change of variables a new viscosity matrix ˜ B has a special block structure. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity u t + f ( u ) x = ( B ( u ) u x ) x . Kawashima Systems: Assumption 3 : Any right eigenvector of f ′ ( u ) is not in the null space of B ( u ) . Assumption 3 , together with Assumption 1 , implies that the system is fully dissipative, even though the viscosity matrix B is rank deficient. Here we use the term full dissipation to indicate the situation where small solutions decay in time, rather than where solutions become regular immediately. Indeed, discontinuities from initial data are permanent in the solution when B is degenerate. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Kawashima-Shizuta Systems: u t + f ( u ) x = ( B ( u ) u x ) x . Perturbation of constant state u 0 = 0 : u ( x , 0 ) ∈ H s , s ≥ 2 . Theorem (Kawashima Dissertation 1984) Golbal existence and decay of solutions. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Hyperbolic characteristics: f ′ ( u ) r j ( u ) = λ j ( u ) r j ( u ) , l j ( u ) f ′ ( u ) = λ j ( u ) l j ( u ) , l j r k = δ jk , j , k = 1 , 2 , · · · , n . Assuming that the p -characteristic is genuinely nonlinear, ∇ λ p ( u ) · r p ( u ) � = 0 , then there exists p -shock waves u ( x , t ) = φ ( x − st ) : − s ( φ − u − ) + f ( φ ) − f ( u − ) = B ( φ ) φ ′ , satisfying the Rankine-Hugoniot condition s ( u + − u − ) = f ( u + ) − f ( u − ) , φ ( ±∞ ) = u ± , and the Lax entropy condition λ p ( u − ) > s > λ p ( u + ) . Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity For a small perturbation of a constant state, say u 0 = 0 , there are accurate approximate solutions the j -diffusion waves θ j ( x , t ) r j ( 0 ) : θ j are self-similar √ θ i = θ i ( x , t ) = ( t + 1 ) − 1 / 2 ζ (( x − λ j ( 0 )( t + 1 )) / t + 1 ) and solutions to θ jt + λ j ( 0 ) θ jx + C ii ( θ 2 j ) x = ( l t j B r j )( 0 ) θ ixx . The solution is unique if we require the total mass to be fixed, � ∞ −∞ θ j ( x , t ) dx = d j . Here the equation is Burgers if the j -characteristic is genuinely nonlinear C jj = 1 2 ( l j ( 0 )) t f ′′ ( 0 )( r j ( 0 ) r j ( 0 )) � = 0 and heat equation if it is linearly degenerate C jj = 0. The viscosity coefficient ( l t j B r j )( 0 ) is postive under Assumption 1 and Assumption 3. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Eigen decompostion of initial perturbation: � ∞ n � u ( x , 0 ) dx = d j r j ( 0 ) . −∞ j = 1 Theorem (Liu-Zeng 1997) n � u ( x , t ) → θ j ( x , t ) r j ( 0 ) , as t → ∞ . j = 1 Proof: Explicit construction of Green’s function for the linearized systems. Duhamel’s principle for pointwise estimates of lower differentials. Kawashima type energy estimates for higher differentials. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Linearized system, around a constant state: u t + A u x = B u xx , A ≡ f ′ ( 0 ) , B ≡ B ( 0 ) . Green’s function (matrix) G ( x , t ) : G t + AG x = BG xx , G ( x , 0 ) = δ ( x ) I . Green’s function G ∗ ( x , t ) for artificial viscosity, µ j ≡ l j B r j > 0 , j = 1 , · · · , n , : H t + diag ( λ 1 , · · · , λ n ) H x = diag ( µ 1 , · · · , µ n ) H xx , H ( x , 0 ) = δ ( x ) I , − ( x − λ 1 t ) 2 √ 4 πµ n t e − ( x − λ nt ) 2 1 1 √ 4 πµ 1 t e H ( x , t ) = diag ( 4 µ 1 t , · · · , ) . 4 µ nt ∗ l 1 . � ∗ H ( x , t ) G ∗ ( x , t ) ≡ � r 1 , · · · , r n . . . l n Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Theorem n m ( x − λ j t ) 2 G ( x , t ) = G ∗ ( x , t )+ O ( 1 )( t + 1 ) − 1 / 2 t − 1 / 2 � e − � + δ ( x − β k t ) P k . 4 Ct j = 1 k = 1 Proof: Fourier transform ˆ G ( η, t ) . Long-Short wave decomposition. Long wave, | η | small, ˆ G ( η, t ) well approximated by ˆ G ∗ , Invert the Fourier transform for the long wave part by complex analytic method to yield heat kernel and faster decaying terms. Short wave, | η | large, expansion to yield δ -functions, intermediate waves by weighted energy and spectral gap. Perturbation theory of Kato. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Perturbation of a p th Lax shock: | u ( x , 0 ) − φ ( x ) | = O ( 1 ) ε ( 1 + | x | ) − 3 / 2 , λ p u − ) > s > λ p ( u + ) . Eigen decomposition � ∞ � � u ( x , 0 ) dx = d j r j ( u − ) + d j r j ( u + ) + x 0 ( u + − u − ) . −∞ j < p j > p Theorem (Liu-Zeng 2013) The solution exists and tends to the combination of diffusion waves and shock wave as t → ∞ : � � u ( x , t ) → φ ( x + x 0 − st ) + θ j ( x , t ) r j ( u − ) + θ j ( x , t ) r j ( u + ) . j < p j > p Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity System of hyperbolic conservation laws: u t + f ( u ) x = 0 . f ′ ( u ) r i ( u ) = λ i ( u ) r i ( u ) , l i ( u ) f ′ ( u ) = λ i ( u ) l i ( u ) . Nonlinear coupling: Hyperbolic nonlinearity C j kl ( u ) ≡ l j ( u ) f ′′ ( u )( r k ( u ) , r l ( u )) , j , k , l = 1 , 2 , · · · , n . The i -th characteristic field genuinely nonlinear iff C i ii � = 0, representing shock forming, [Lax]. C j kl , j � = k or j � = k , measure nonlinear inviscid coupling. l j B r k , coupling due to viscosity. The nonlinear coupling yields nonlinear wave interactions and the wave patterns of the following kinds: Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity Coupling waves: i ( t + 1 )) 2 + t + 1 ] − 1 ψ i ( x , t ) ≡ [( x − λ 0 2 , i � = p , i ( t + 1 ) | 3 + ( t + 1 ) 2 ] − 1 ¯ ψ i ( x , t ) ≡ [ | x − λ 0 3 , i � = p , 3 1 χ i ( x , t ) ≡ min { ε − 1 1 2 ( t + 1 ) − 1 2 ψ 2 ψ i ( x , t ) , ε 2 i ( x , t ) } char i ( x , t ) , 2 i � = p , 1 , if 0 < x < λ 0 i ( t + 1 ) and i > p 1 , if λ 0 char i ( x , t ) ≡ i ( t + 1 ) < x < 0 and i < p , 0 , otherwise ψ p ( x , t ) ≡ [( | x | + ε ( t + 1 )) 2 + t + 1 ] − 1 2 , ψ p ( x , t ) ≡ [( | x | + ε ( t + 1 )) 3 + ( t + 1 ) 2 ] − 1 ¯ 3 , η ( x , t ) ≡ min { ε ( x 2 + 1 ) − 1 2 ( | x | + t + 1 ) − 1 2 , ( x 2 + 1 ) − 3 4 ( | x | + t + 1 ) − 1 2 } . Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
Physical Viscosity For the combined energy estimates (for higher order differentiations) and pointwise estimates (for lower order differentiations) we require that the initial perturbation U ( x , 0 ) = u ( x , 0 ) − φ ( x + x 0 ) ∈ H 13 ( R ) . To effectviely describe the wave coupling, we further require the pointwise decay of the perturbation. Thus we assume that 3 ∂ j ∂ 4 � � � � � 3 5 � � � � � δ 0 ≡ sup ( | x | + 1 ) ∂ x j U ( x , 0 ) � + ( | x | + 1 ) ∂ x 4 U ( x , 0 ) 2 4 � � � � � � � x ∈ R j = 0 6 8 ∂ j ∂ j � � � � � 1 � � � � � � +( | x | + 1 ) ∂ x j U ( x , 0 ) � +( | x | + 1 ) ∂ x j U ( x , 0 ) + � U ( x , 0 ) � H 13 2 � � � � � � � j = 5 j = 7 is bounded. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity
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