Velocity averaging, kinetic formulations and regularizing effects in conservation laws and related PDEs. Eitan Tadmor. University of Maryland

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1 Velocity averaging, kinetic formulations and regularizing effects in conservation laws and related PDEs Eitan Tadmor Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics, Institute for Physical Science & Technology University of Maryland tadmor

2 Loss of regularity Nonlinear conservation law: t ρ + x A(ρ) = 0, a(ρ) := A (ρ) Const. C 1 initial data ρ(x, 0) = ρ 0 (x) Shock discontinuities: C 1 -smoothness breakdown at a finite time t = t c. Generic: t c 1/ x a(ρ 0 (x)) > 0

3 Gain of regularity: t ρ + x A(ρ) = 0, a(ρ) := A (ρ) Nonlinearity: A(ρ) is convex A ( ) = a ( ) α > 0 t ρ 1 at (x 1, t) ρ 2 at (x 2, t) y 1 = x 1 ta(ρ 1 ) y 2 = x 2 ta(ρ 2 ) Entropy solution: y 2 y 1 x 2 x t a(ρ 2) a(ρ 1 ) x 2 x 1 α ρ 2 ρ 1 x 2 x 1 Olienik condition: ρ x (t, x) αt 1 ; dispersive effect Regularizing effect: S(t) : L BV ; compactness, irreversibility,... Compensated compactness: Tartar a (ρ) 0 a.e.

4 Analysis of nonlinear conservation laws BV compactness translation invariance; mostly 1D Compensated compactness restricted to 1+1 D; but... t ρ + x1 A 1 (ρ) + x2 A 2 (ρ) = 0 Nonlinearity: (E & Engquist, Bagnerini, Rascle & ET): If meas {v ξ 1 a 1 (v) + ξ 2 a 2 (v) = 0} = 0 = S(t) is compact Measure valued solutions restricted to scalar eq s Averaging velocity lemma kinetic formulation

5 Plan of talk Kinetic formulation Averaging lemma for first and second order eq s Regularizing effects: nonlinear conservation laws convection-(degenerate) diffusion elliptic eq s Joint work w/t. Tao

6 Second-order eq s and their kinetic formulations d j=0 xj A j (ρ) d j,k=1 2 x j x k B jk (ρ) = S(ρ) Conservation laws: A 0 (ρ) = ρ, B jk 0 Degenerate diffusion, convection-diffusion: {B jk } 0 Elliptic: A j 0 On a proper notion of a (weak) solution Kinetic formulation: pseudo-maxwellian χ ρ (v) := +1 0 < v < ρ 1 ρ < v < 0 0 otherwise macroscopic ρ(x) = average of its microscopic distribution: ρ = velocity averaging: A(ρ) = a(v) x χ ρ v a(v)χ ρ(v)dv, B(ρ) = v χ ρ(v)dv v b(v)χ ρ(v)dv: b(v) x, x χ ρ + S(v) v χ ρ =: v g(x, v) RHS Entropy/renormalized...solutions: RHS = v m(x, v), m M +

7 Entropy/renormalized...solutions d j=0 xj A j (ρ) d j,k=1 2 x j x k B jk (ρ) S(ρ) = RHS=0 {}}{ ɛ ρ ɛ 0

8 Entropy/renormalized...solutions η (ρ ɛ ), d j=0 xj A j (ρ ɛ ) d j,k=1 x 2 j x k B jk (ρ ɛ ) S(ρ ɛ ) = η (ρ ɛ ), ɛ ρ ɛ ɛ 0 ( ) η a j =: A η j {}}{ d j=0 xj A η j (ρ) ( ) η b jk =: B η jk {}}{ d j,k=1 2 x j x k B η jk (ρ) η (ρ)s(ρ) negative measure if η >0 {}}{ = ɛη (ρ) ρ 0 v g(x, v) = a(v) x χ ρ b(v) x, x χρ + S(v) v χ ρ

9 η (ρ ɛ ), d j=0 ( ) η a j =: A η j {}}{ d j=0 xj A η j (ρ) Entropy/renormalized...solutions xj A j (ρ ɛ ) d j,k=1 ( ) η b jk =: B η jk {}}{ d j,k=1 x 2 j x k B jk (ρ ɛ ) S(ρ ɛ ) 2 x j x k B η jk (ρ) η (ρ)s(ρ) = η (ρ ɛ ), ɛ ρ ɛ ɛ 0 negative measure if η >0 {}}{ = ɛη (ρ) ρ 0 η (v) v g(x, v) = η (v)a(v) x χ ρ η (v)b(v) x, x χρ +η (v)s(v) v χ ρ for all convex η s: η (v) v g(v)dv = j x j A j (ρ) {}}{ η (v)a(v) x χ ρ j,k 2 x j x k B η j,k (ρ) {}}{ η (v)b(v) x, x χρ + η (ρ)s(ρ) {}}{ η (v)s(v) v χ ρ 0 = g(v) is a nonnegative measure: g(v) = m(v) M + measures entropy dissipation; supported on shocks

10 Kinetic examples: ρ(x, v) f(x, v) = χ ρ(x) (v) Scalar conservation laws: ρ t + x A(ρ) = 0, a( ) := A ( ) Kinetic formulation Lions, Perthame, ET: ( ) t + a(v) x f(x, t, v) = v m(x, t, v), m M + Convection-(degenerate) diffusion equations: ρ t + x A(ρ) tr { x x B(ρ)} = v m(x, t, v), b( ) := B ( ) 0 Kinetic formulation (no uniqueness) ( t + a(v) x b(v) x, x ) f(x, t, v) = v m(x, t, v), m M + Renormalized solution of Chen-Perthame, Souganidis, Karlsen,... Elliptic eq s: tr { x x B(ρ)} = S(ρ), b 0, sgn(c)s(c) 0,... Kinetic formulation: ( b(v) x, x + S(v) v ) f(x, v) = v m(x, v), m M +

11 Velocity Averaging: 1/3 ( a(v) x b(v) x, x ) χ ρ(x) (v) = v m(x, v) L k ( x, v)f(x, v) = RHS(x, v), k = 1, 2,..., Hyperbolic regularity: f(x, v) is as smooth as RHS(x, v) Averaging lemma: f(x) := f(x, v)dv is smoother Averaging in the microscopic scale v: leads to gain of smoothness in the macroscopic scale x First-order L s: Agoshkov, Perthame, Golse, Lions, Gerard,... DiPerna-Lions-Meyer Jabin, DeVore-Petrova,..., Golse-Saint-Raymond, Liu-Yu mixture lemma, Panov et. al.,... Second-order L s... Brenier, Lions-Perthame-ET, Souganidis,...

12 Velocity averaging 2/3: L k ( x, v)f = RHS(x, v) ˆf(ξ, v) = Non-degeneracy Ω δ (ξ) := RHS(ξ, v) L k (ξ, v) : gain of regularity whenever L k(ξ, v) 0: { v : L k (ξ, v) < δ} Ωδ (ξ) < ( δ J β ) α, ξ J f(x) ξ J F 1 v 1 Ω δ (ξ) (v) ˆf(ξ, v)dv }{{} + ξ J F 1 1 Ω v c δ (ξ)(v) RHS(ξ, v) L k (ξ, v) dv }{{} = b J (x) + g J (x)

13 Velocity averaging 2/3: L k ( x, v)f = RHS(x, v) ˆf(ξ, v) = Non-degeneracy Ω δ (ξ) := RHS(ξ, v) : gain of regularity whenever L(ξ, v) 0: L k (ξ, v) { v : L k (ξ, v) < δ} Ωδ (ξ) < ( δ J β ) α, ξ J f(x) ξ J 1 Ωδ (ξ) (v) ˆf(ξ, v)dv F 1 } v {{} + ξ J 1 Ω c δ (ξ) (v) RHS(ξ, v) L k (ξ, v) dv F 1 v }{{} = b J (x) + g J (x) Key point: F 1 v ϕ( L(ξ, v) ) ˆf(ξ, v)dv :L p x δ Lp x independent of δ b J L p < f L p (x,v) ( ) α δ p ( ) α δ J β, g J W k (L q ) < RHS L q q 1 (x,v) J β f L p, RHS L q = ρ = f W s (L 1 ), s = s(α, β, p, q) > 0

14 Velocity averaging: 3/3 L( x, v)f(x, v) = RHS; deg(l(ξ, )) k, f χ ρ, RHS v m L(ξ, ) The truncation property: ϕ( ) : L p x δ Lp x independent of δ Non-degeneracy: meas v {v L(ξ, v) < δ } Const. ( δ J β ) α, ξ J f L p p=2, RHS = vm, m L q q=1 f W s (L 1 ), s < (2β k)α 2α + 1 If deg(l(ξ, )) = k 2 then it satisfies the truncation property 1st-order: L(ξ, v) = a(v) iξ (β = k = 1), f W s (L 1 ), s α = α 2α+1 2nd-order: L(ξ, v) = b(v)ξ, ξ (β = k = 2), f W s (L 1 ), s α = 2α 2α+1

15 Nonlinear conservation laws 1/2 (Lions-Perthame-ET) Kruzkov s entropy pairs: ρ t + x A(ρ) = 0 η(ρ; v) = ρ v, A η j = sgn(ρ v)( A j (ρ) A j (v) ) : m(t, x, v) := Differentiate: [ t η(ρ; v) η(0; v) 2 + x ( A η (ρ; v) A η (0; v) t χ ρ(t,x) (v)+a(v) x χ ρ(t,x) (v) = v m(t, x, v) in D ((0, ) IR d x IR v) 2 ) ] Nonlinearity: meas v {v A (v) ξ < δ } < δ α, ξ = 1 Averaging: χ ρ L 2, m M: = ρ(x, t) = χ ρ(x,t) (v)dv W s α (L 1 ), s α = α 2α+1

16 Nonlinear conservation laws 2/2 ρ t + x A(ρ) = 0, meas v {v A (v) ξ < δ } < δ α ρ W s,1, s = α 2α + 1 Examples of regularizing effects: ρ W s (L 1 ): t ρ + (sin(ρ)) x1 + ( 1 3 ρ3 ) x2 = 0 s < 1 6 τ + ξ 1 cos(v) + ξ 2 v 2 < δ α = 1/4 t ρ + ( ρ m ρ) x1 + ( ρ l ρ) x2 = 0 s < min { 1 m + 2, 1 }, m l l + 2

17 Nonlinear conservation laws 2/2 (continued) Propagation of oscillations: Lack of compactness: 2D Burgers ρ t + ( ρ2 2 ) x 1 + ( ρ2 2 ) x 2 = 0: linearized symbol ξ 0 + ξ 1 v + ξ 2 v = 0, ξ 1 + ξ 2 = 0, ξ 0 = 0 ρ 0 (x y) are steady solution - oscillations persist along x 1 x 2 = const. Not BV optimal; but 1 3 optimality w/α = 1 - De Lellis & M. Westdickenberg

18 Nonlinear parabolic equations 1/4 (ET-Tao) t ρ + x A(ρ) 2 x j x k B jk (ρ) = 0, a := A, b := B 0 Kinetic formulation for f = χ ρ : t f + a(v) x f b(v) x, x f = v m, m = m A + m B M + Non-degeneracy: { v : τ + a(v) ξ + b(v)ξ, ξ < δ} < δ α Parabolic degeneracy... λ 1 (v) λ 2 (v)... λ N (v) 0 { v : λ N (v) δ } < δ α = ρ(x, t) W 2α 2α+1(L 1 ), t ɛ Porous media: ρ t + (ρ n+1 ) = 0, b(v) v n ρ W 2 n+2(l 1 ) Beyond the isotropic cases...

19 t Nonlinear Convection diffusion 2/4 (ET-Tao) { } 1 ρ(t, x) + x l + 1 ρl+1 (t, x) 2 { } 1 x 2 n + 1 ρn+1 (t, x)) = 0, has a regularizing effect of order s: t ɛ : ρ 0 L (IR x ) ρ(t, ) W s,1 loc (IR x) If n l then degenerate diffusion dominates: s = 2 n + 2 If n 2l then nonlinear hyperbolicity dominates: s = 1 l + 2 Intermediate regularization l < n < 2l: s = n + (2l n)ζ 2n + 2(l n)ζ + nl, ζ = n l 1

20 Nonlinear parabolic equations 2/4 (continued) Study the degenerate set ω L (J; δ) := Ω L (J; δ), Ω L (J; δ) := { v : L(τ, ξ, v) δ, τ 2 + ξ 2 J } Ω L ( J; δ ) := {v J τ + vl ξ + J 2 v n ξ 2 δ }, τ 2 +ξ 2 = 1, J > 1, δ < 1. case #1: dominated by the diffusive part, when n l: Ω L (J; δ) Ω b := { v : v n δ/j 2} ω L (J; δ) < (δ/j 2 ) 1/n α = 1/n, β = 2 case #2: dominated by the convective part, when n 2l: Ω L (J; δ) Ω a := {v : v l < δ/j} ω L (J; δ) < (δ/j) 1/l α = 1/l, β = 1

21 Nonlinear parabolic equations 2/4 (final) intermediate cases #3: l < n < 2l interpolate... Bootstrap argument ρ W θ (L 1 ): ρ W s,1 = χ ρ W s,1 L = χ ρ W s,2, s (1 θ) s 2 + kθ ρ W s,1, s = 2kθ 1 + θ L p -initial data (nonlinear interpolation w/l 1 -contraction) S(t) : ρ 0 L p L 1 W s,1, s < kα (2α + 1)p, p > 1

22 Non-isotropic diffusion 3/4 (ET-Tao) ρ(t, x) + t x 1 { } 1 l + 1 ρl+1 (t, x) + { } 1 x 2 m + 1 ρm+1 (t, x) 2 j,k=1 with non-degenerate diffusion, B (v) > v n : 2 x j x k B jk (ρ(t, x)) = 0 W s,1 -regularity, t ɛ : ρ 0 L (IR x ) ρ(t, ) W s,1 loc (IR x): s < min { 1 l + 2, 1 m n + 2, } if n > 2 max(l, m) + 1 and l m if n < min(l, m) or l = m

23 Fully-degenerate case 4/4 (ET-Tao) ρ(t, x) + t ( + ){ } 1 x 1 x 2 l + 1 ρl+1 (t, x) ){ 1 x 1 x 2 n + 1 ρn+1 (t, x) ( 2 x 2 1 x 2 2 } = 0, lack of regularity on hyperbolic and parabolic parts: ρ(t, x) t ( ρ(t, x) + + t x 1 x 2 ( x 1 x 2 x 2 1 x 2 2 ) ρ = 0 : ρ 0 (x y) ) ρ = 0 : ρ 0 (x + y) Full equation admits W s,1 -regularizing effect of order s = s(l, n): if n > 2l : s(l, n) = n l

24 Degenerate elliptic equations 2 x j x k B jk (ρ) = S(ρ), b := B 0, sgn(c)s(c) C 0 Kinetic formulation for f = χ ρ : ( b(v) x, x + S(v) v ) f = v m, m M + Non-degeneracy: Ω δ < δ α, Ω δ := { } v : b(v)ξ, ξ δ Interior regularity of L -solution ρ D (G): ρ(x) W s,1 loc (G 1), s < min ( α, 2α 2α + 1 ) Beyond the isotropic cases...

25 Final notes Nonlinear conservation laws Convergence on unstructured grids: L 1 contraction w/no BV bound (Noelle, Westdickenberg,...) Compactness and nonlinearity in multid case: Compensated compactness vs. kinetic formulation Convergence towards steady state: (E, Engquist,...) Entropy/kinetic solution of elliptic equations Improved regularity: the first order case (s> 1/3) Improved regularity: the second-oder dissipation What about equations of order> 2? Systems: multivalued solutions, (Lions, Perthame, Souganidis, ET, Brenier,...)

26 THANK YOU