On variational kinetic formulations for scalar conservation laws and - PowerPoint PPT Presentation

On variational kinetic formulations for scalar conservation laws and the Euler equations of gas dynamics. Misha Perepelitsa University of Houston, Houston, USA HYP2012, Padova, Italia

Content: 1. Scalar conservation laws: ( x, t ) ∈ R n +1 (S.C.L.) ∂ t u + div f ( u ) = 0 , , + f ∈ C 1 ( R ) n . ◮ Kinetic formulation (Lions-Perthame-Tadmor). ◮ Variational kinetic formulation (Panov, Brenier). 2. Kinetic formulations for the Euler equations: ρ t + div ( ρu ) = 0 , (E.eqs.) ( ρu ) t + div ( ρu ⊗ u ) + ∇ p = 0 , ( ρE ) t + div ( ρEu + pu ) = 0 , ρ – density, u = ( u 1 , ..., u n ) – velocity, E = | u | 2 + e, -total energy, e – 2 internal energy, p = ( γ − 1) ρe = RρT.

(S.C.L.) ∂ t u + div f ( u ) = 0 , u ( t = 0) = u 0 . Entropy-entropy flux pair ( η, q ) : q ′ ( u ) = f ′ ( u ) η ′ ( u ) . u ( x, t ) is an entropy solution if for any convex entropy-entropy flux pair ( η, q ) , D ′ ( R n +1 ∂ t η ( u ) + div q ( u ) ≤ 0 , ) . + (Kruzhkov) For any u 0 ∈ L ∞ ( R n ) , there a unique entropy solution of (S.C.L.) and u ∈ C ([0 , + ∞ ); L 1 loc ( R n )) . For any two entropy solutions u, v with the data u 0 , v 0 ∈ L ∞ ∩ L 1 ( R n ) , 1. for all t > 0 , Z Z | u ( x, t ) − v ( x, t ) | dx ≤ | u 0 ( x ) − v 0 ( x ) | dx ; 2. if u 0 ≤ v 0 a.e. R n , a.e. R n +1 u ( x, t ) ≤ v ( x, t ) , . +

(S.C.L.) ∂ t u + div f ( u ) = 0 , u ( t = 0) = u 0 . Entropy-entropy flux pair ( η, q ) : q ′ ( u ) = f ′ ( u ) η ′ ( u ) . u ( x, t ) is an entropy solution if for any convex entropy-entropy flux pair ( η, q ) , D ′ ( R n +1 ∂ t η ( u ) + div q ( u ) ≤ 0 , ) . + (Kruzhkov) For any u 0 ∈ L ∞ ( R n ) , there a unique entropy solution of (S.C.L.) and u ∈ C ([0 , + ∞ ); L 1 loc ( R n )) . For any two entropy solutions u, v with the data u 0 , v 0 ∈ L ∞ ∩ L 1 ( R n ) , 1. for all t > 0 , Z Z | u ( x, t ) − v ( x, t ) | dx ≤ | u 0 ( x ) − v 0 ( x ) | dx ; 2. if u 0 ≤ v 0 a.e. R n , a.e. R n +1 u ( x, t ) ≤ v ( x, t ) , . + We will assume that u ( x, t ) is L –periodic in x and for some M > 0 , 0 < essinf u ≤ esssup u < M.

Smooth solutions. For the initial data u 0 , choose a level set function Y 0 ( x, v ) : Y 0 ( x, u 0 ( x )) = λ. Consider  ∂ t Y + f ′ ( v ) · ∇ x Y = 0 , Y ( t = 0) = Y 0 ( x, v ) . For all times t ∈ (0 , t ∗ ) while there is u ( x, t ) such that { ( x, v ) : Y ( x, t, v ) = λ } = { ( x, u ( x, t )) } , u ( x, t ) is a classical solution of (S.C.L.).

Smooth solutions. For the initial data u 0 , choose a level set function Y 0 ( x, v ) : Y 0 ( x, u 0 ( x )) = λ. Consider  ∂ t Y + f ′ ( v ) · ∇ x Y = 0 , Y ( t = 0) = Y 0 ( x, v ) . For all times t ∈ (0 , t ∗ ) while there is u ( x, t ) such that { ( x, v ) : Y ( x, t, v ) = λ } = { ( x, u ( x, t )) } , u ( x, t ) is a classical solution of (S.C.L.). multi-valued solution = length of dashed intervals

Averaging of multi-valued solutions.  0 v < u 0 ( x ) Let Y 0 ( x, v ) = , 1 v ≥ u 0 ( x ) Z + ∞ u ∗ ( h, x ) = (1 − Y 0 ( x − f ′ ( v ) h, v )) dv. 0 Let ω ( x ) be a test function and compute Z ( u ∗ ( h, x ) − u 0 ( x )) ω ( x ) dx Z Z + ∞ (1 − Y 0 ( x − f ′ ( v ) h, v )) − (1 − Y 0 ( x, v )) ˆ ˜ = ω dxdv 0 Z + ∞ Z (1 − Y 0 ( x, v ))( ω ( x + f ′ ( v ) h ) − ω ( x )) dxdv = 0 Z f ( u 0 ( x )) ω x dx + O ( h 2 ) . = h u ∗ is approximately a weak solution of (S.C.L.) on t ∈ [0 , h ] .

Time discrete BGK-type approximation (Brenier, Giga-Miyakawa): Define the kinetic function as (for u > 0):  0 v < u Y ( v, u ) = 1 v ≥ u. Y ( v, u ( x )) – kenetic density of u ( x ) . Let h > 0 – time step, n ∈ N , ◮ Given u n − 1 ( x ) , set Y n − 1 ( x, v ) = Y ( v, u n − 1 ( x )) , solve 8 ∂ t Y + f ′ ( v ) · ∇ x Y = 0 , < Y ( x, 0 , v ) = Y n − 1 ( x, v ) , : and define Z M u n ( x, v ) = Y n ( x, v ) = Y ( v, u n ( x )) . (1 − Y ( x, h, v )) dv, 0

Time discrete BGK-type approximation (Brenier, Giga-Miyakawa): Define the kinetic function as (for u > 0):  0 v < u Y ( v, u ) = 1 v ≥ u. Y ( v, u ( x )) – kenetic density of u ( x ) . Let h > 0 – time step, n ∈ N , ◮ Given u n − 1 ( x ) , set Y n − 1 ( x, v ) = Y ( v, u n − 1 ( x )) , solve 8 ∂ t Y + f ′ ( v ) · ∇ x Y = 0 , < Y ( x, 0 , v ) = Y n − 1 ( x, v ) , : and define Z M u n ( x, v ) = Y n ( x, v ) = Y ( v, u n ( x )) . (1 − Y ( x, h, v )) dv, 0 ◮ Setting u n = S h ( u n − 1 ) : ◮ � S h ( u ) − S h ( v ) � L 1 ≤ � u − v � L 1 ; ◮ � S h 1 ( u ) − S h 2 ( u ) � L 1 ≤ C | h 1 − h 2 | TV ( u ) .

Time discrete BGK-type approximation (Brenier, Giga-Miyakawa): Define the kinetic function as (for u > 0):  0 v < u Y ( v, u ) = 1 v ≥ u. Y ( v, u ( x )) – kenetic density of u ( x ) . Let h > 0 – time step, n ∈ N , ◮ Given u n − 1 ( x ) , set Y n − 1 ( x, v ) = Y ( v, u n − 1 ( x )) , solve 8 ∂ t Y + f ′ ( v ) · ∇ x Y = 0 , < Y ( x, 0 , v ) = Y n − 1 ( x, v ) , : and define Z M u n ( x, v ) = Y n ( x, v ) = Y ( v, u n ( x )) . (1 − Y ( x, h, v )) dv, 0 ◮ Set u h ( x, hn ) = u n ( x ) and linearly interpolate for t ∈ ( h ( n − 1) , hn ) . Then u h → u in C ([0 , T ); L 1 ( R n )) , ∀ T > 0 , and u is a solution of (S.C.L.).

Time discrete BGK-type approximation (Brenier, Giga-Miyakawa): Define the kinetic function as (for u > 0):  0 v < u Y ( v, u ) = 1 v ≥ u. Y ( v, u ( x )) – kenetic density of u ( x ) . Let h > 0 – time step, n ∈ N , ◮ Given u n − 1 ( x ) , set Y n − 1 ( x, v ) = Y ( v, u n − 1 ( x )) , solve 8 ∂ t Y + f ′ ( v ) · ∇ x Y = 0 , < Y ( x, 0 , v ) = Y n − 1 ( x, v ) , : and define Z M u n ( x, v ) = Y n ( x, v ) = Y ( v, u n ( x )) . (1 − Y ( x, h, v )) dv, 0 ◮ Set u h ( x, hn ) = u n ( x ) and linearly interpolate for t ∈ ( h ( n − 1) , hn ) . Then u h → u in C ([0 , T ); L 1 ( R n )) , ∀ T > 0 , and u is a solution of (S.C.L.). ◮ (Vasseur) Convergence without BV bounds.

(Perthame-Tadmor) Continuous time BGK-type approximation. ∂ t Y + f ′ ( v ) · ∇ x Y = ε − 1 ( Y ( v, u ( x, t )) − Y ) , R M u ( x, t ) = 0 (1 − Y ( x, t, v )) dv. ◮ u ε → u – solution of (S.C.L.).

(Perthame-Tadmor) Continuous time BGK-type approximation. ∂ t Y + f ′ ( v ) · ∇ x Y = ε − 1 ( Y ( v, u ( x, t )) − Y ) , R M u ( x, t ) = 0 (1 − Y ( x, t, v )) dv. ◮ u ε → u – solution of (S.C.L.). Kinetic formulation of Lions-Perthame-Tadmor. u ( x, t ) is an entropy solution of (S.C.L.) iff there is a nonnegative measure m ∈ M + ( R n +2 ) , + and Y ( x, t, v ) = Y ( v, u ( x, t )) solves: ∂ t Y + f ′ ( v ) · ∇ x Y = − ∂ v m, (K.eq.) Y ( x, 0 , v ) = Y ( v, u 0 ( x )) . Applications: t,x , s ∈ (0 , 1 / 3), -regularity of L ∞ solutions. (Lions-Perthame-Tadmor) W s, 1 (De Lellis-Otto-Westdickenberg) Structure of L ∞ solutions.

Measure-valued solutions. Let for every ( x, t ) , Y ( x, t, v ) be non-decreasing in v and Y ( x, t, 0) = 0 , Y ( x, t, M ) = 1 and ∂ t Y + f ′ ( v ) · ∇ x Y = − ∂ v m, m ∈ M + ( R n +2 ) . + ◮ Y ( x, t, v ) defines a probability measure ν x,t on R : ν x,t (( v 1 , v 2 ]) = Y ( x, t, v 2 ) − Y ( x, t, v 1 ) , and for any convex entropy-entropy flux pair ( η, q ) : D ′ ( R n +1 ) . ∂ t � η, ν x,t � + ∂ x � q, ν x,t � ≤ 0 ,

Measure-valued solutions. Let for every ( x, t ) , Y ( x, t, v ) be non-decreasing in v and Y ( x, t, 0) = 0 , Y ( x, t, M ) = 1 and ∂ t Y + f ′ ( v ) · ∇ x Y = − ∂ v m, m ∈ M + ( R n +2 ) . + ◮ Y ( x, t, v ) defines a probability measure ν x,t on R : ν x,t (( v 1 , v 2 ]) = Y ( x, t, v 2 ) − Y ( x, t, v 1 ) , and for any convex entropy-entropy flux pair ( η, q ) : D ′ ( R n +1 ) . ∂ t � η, ν x,t � + ∂ x � q, ν x,t � ≤ 0 , ◮ (Tartar) Compensated compactness method. ◮ (Schochet) Entropy mv-solutions (with given ν 0 ,x ) are not unique. ◮ (DiPerna) MV-solutions with ν 0 ,x = δ u 0 ( x ) , coincide with weak entropy solutions.

Measure-valued solutions. Let for every ( x, t ) , Y ( x, t, v ) be non-decreasing in v and Y ( x, t, 0) = 0 , Y ( x, t, M ) = 1 and ∂ t Y + f ′ ( v ) · ∇ x Y = − ∂ v m, m ∈ M + ( R n +2 ) . + ◮ Y ( x, t, v ) defines a probability measure ν x,t on R : ν x,t (( v 1 , v 2 ]) = Y ( x, t, v 2 ) − Y ( x, t, v 1 ) , and for any convex entropy-entropy flux pair ( η, q ) : D ′ ( R n +1 ) . ∂ t � η, ν x,t � + ∂ x � q, ν x,t � ≤ 0 , ◮ (Tartar) Compensated compactness method. ◮ (Schochet) Entropy mv-solutions (with given ν 0 ,x ) are not unique. ◮ Take Y ( x, 0 , v ) = v/M, independent of x. ◮ Take m 1 ( x, t, v ) ≡ 0 and m 2 ( x, t, v ) = m ( v ) ≥ 0 , m ′ (0) = m ′ ( M ) = 0 . ◮ Obtain two solutions v/M and v/M − tm ′ ( v ) . ◮ (DiPerna) MV-solutions with ν 0 ,x = δ u 0 ( x ) , coincide with weak entropy solutions.

Variational property of the kinetic solutions Y = Y ( v, u ( x, t )) . ◮ Z L Z M Z L d Y 2 dxdv = − d u ( x, t ) dx = 0 . dt dt 0 0 0 Consider ∂ t Y + f ′ ( v ) · ∂ x Y = − ∂ v m. ◮ Let ˜ Y ( x, v ) be non-decreasing in v test function, then Z L Z M ( ˜ Y − Y )( ∂ t Y + f ′ ( v ) · ∂ x Y ) dxdv ≥ 0 . (V.K.eq.) 0 0 ◮ (V.K.eq.) is equivalent to (K.eq.) if Y = Y ( x, u ( x, t )) , but more restrictive when Y comes from mv-solution. ◮ For mv-solutions (V.K.eq.) imposes a non-linear constraint Z L Z M d Y 2 dxdv = 0 . dt 0 0