Lipschitz Metrics for Non-smooth evolutions

Transkript

1 Lipschitz Metrics for Non-smooth evolutions Alberto Bressan Department of Mathematics, Penn State University Alberto Bressan (Penn State) Nonsmooth evolutions 1 / 36

2 Well-posedness for evolution equations d u(t) = A u(t) dt d dt u(t) v(t) κ u(t) v(t) (P1) = u(t) v(t) e κt u(0) v(0) (P) How can we prove well-posedness of the Cauchy problem if (P1) fails? Construct some other weighted distance d (, ) for which (P1) or (P) hold Alberto Bressan (Penn State) Nonsmooth evolutions / 36

3 Toy model 1: A discontinuous O.D.E. ẋ = f () = { 1 if < 0, if 0, 1 (t) (t) 1 (0) (0) The flow is an isometry w.r.t. the weighted distance d (, y) = y if, y 0 y if, y 0 + y if < 0 < y (Similar to hyperbolic systems of conservation laws) Alberto Bressan (Penn State) Nonsmooth evolutions 3 / 36

4 (t) 1 (t) (t) 1 (t) ε ε ε t ε t Alberto Bressan (Penn State) Nonsmooth evolutions 4 / 36

5 Toy model ẋ = (1) Definition: a solution is admissible iff it is strictly increasing For every initial data (0) = 0, the Cauchy problem has a unique admissible solution, depending continuously on the initial data The estimates (P1)-(P) both fail, for the usual distance d(, y) = y The flow is an isometry w.r.t. the weighted distance d (, y) = y 1 ds s (Similar to Hunter-Saton, Camassa-Holm, variational wave equation...) Alberto Bressan (Penn State) Nonsmooth evolutions 5 / 36

6 An error estimate Theorem. Let S : X [0, [ X be a Lipschitz semigroup on a metric space (X ; d) satisfying d(s tu, S sv) L d(u, v) + L t s u, v X, t, s [0, T ] Then, for every Lipschitz continuous map w : [0, T ] X one has ( ) ( ) T d w(t + h), S h w(t) d w(t ), S w(0) L lim inf T 0 h 0+ h dt = L T 0 [instantaneous error rate at time t] dt w(t) w(t+h) w(t) S h w(t) S w(t+h) T t h S w(t) T t S w(0) T w(0) Alberto Bressan (Penn State) Nonsmooth evolutions 6 / 36

7 Riemann distance Length of a path γ : [0, 1] M on a Riemann manifold M γ(s) γ (s) b = γ (1) γ = 1 g ij (γ(s)) γ i (s) γ j (s) ds 0 ij a = γ (0) g = g ij = Riemann tensor d (a, b) = { } inf γ ; γ(0) = a, γ(1) = b Toy problem 1: g() = { 1 if < 0 1/ if > 0 Toy problem : g() = 1 Alberto Bressan (Penn State) Nonsmooth evolutions 7 / 36

8 Finsler metrics Weighted length of a path γ : [0, 1] M in an infinite dimensional manifold: γ(s) b = γ (1) γ = 1 0 γ(s) γ(s) ds a = γ (0) γ (s) T u = tangent space at u M is a Banach space with norm u. For each s [0, 1], the vector γ(s) T γ(s) has norm γ(s) γ(s) Finsler distance = length of shortest path { } d (a, b) = inf γ ; γ(0) = a, γ(1) = b Alberto Bressan (Penn State) Nonsmooth evolutions 8 / 36

9 Hyperbolic Systems of Conservation Laws u t + f (u) = 0 u = (u 1,..., u n ) R n conserved quantities f = (f 1,..., f n ) : R n R n flues u t + A(u)u = 0 A(u) = Df (u) STRICTLY HYPERBOLIC: each matri A(u) has real distinct eigenvalues λ 1 (u) < λ (u) < < λ n (u) a basis of eigenvectors r 1 (u),..., r n (u) Alberto Bressan (Penn State) Nonsmooth evolutions 9 / 36

10 (P1) d u(t) ũ(t) L 1 κ u(t) ũ(t) L 1 FAILS! dt shocks in u shocks in u ~ t τ+ h τ u ~ u L 1 τ τ+h t (P3) u(t) ũ(t) L 1 C u(t) ũ(t) L 1 HOLDS for solutions with small total variation Alberto Bressan (Penn State) Nonsmooth evolutions 10 / 36

11 A space of generalized tangent vectors εv εξ α u ε u α + α εξ α u piecewise Lipschitz with jumps at points 1 < < < N First order perturbation: u ε () = u() + εv() [u( α+) u( α )] χ [α, α+εξ α] + [u( α+) u( α )] χ [α+εξα, ξ α] ξ α>0 ξ α<0 Tangent vector: (v ; ξ 1,..., ξ n ) T u = L 1 (R) R N v = vertical displacement, ξ α = shift in the jump at α Alberto Bressan (Penn State) Nonsmooth evolutions 11 / 36

12 A weigthed metric on the tangent space εv εξ α u ε u α + α εξ α Standard ( L 1 ) norm on tangent vectors Weighted norm: (v ; ξ 1,..., ξ N ) =. v L 1 +. n (v ; ξ 1,..., ξ n) = i=1 N u( α+) u( α ) ξ α. α=1 W u i () v i () d + v i = i-th component of v along a basis of eigenvectors r 1(u),..., r n(u) N Wk u α () u( α+) u( α ) ξ α. α=1 Assume: u has a shock at α in the family k α Alberto Bressan (Penn State) Nonsmooth evolutions 1 / 36

13 W u i () = weight given to an i-wave in u( ) at W u i () = 1 + κ 1 A i () + κ Q A i () = total strengths of all waves approaching an i-wave at Q = Glimm interaction potential, 1 << κ 1 << κ Main steps in the analysis: derive a linearized evolution equation for a tangent vector (v, ξ) d (assuming enough regularity) prove that (v(t), ξ(t)) 0 dt u(t) conclude that the semigroup generated by the hyperbolic system of conservation laws is contractive w.r.t. the equivalent distance d (u, ũ) = inf{ γ ; γ(0) = u, γ(1) = ũ} Alberto Bressan (Penn State) Nonsmooth evolutions 13 / 36

14 u(0) ~ w (0) θ u (0) u(0) θ u(t) ~ γ t u(t) θ w (t) u ~ (T) u(t) w (T) θ θ u (T) d Assume dt (v, ξ)(t) u(t) 0 for every solution u( ) and every corresponding tangent vector (v, ξ). Let γ t : θ u θ (t) be a path in L 1, with tangent vectors d dθ uθ (t) = w θ (t) = (v θ (t), ξ θ (t)) T u(t) Then the path length γ t = 1 0 (v θ, ξ θ )(t) u θ (t) dθ does not increase in time Alberto Bressan (Penn State) Nonsmooth evolutions 14 / 36

15 The Camassa-Holm Equation ( ) u (CH) u t + + P = 0 P =. 1 ( ) e u + u ( ) u t ( ) u u P = u P ( ) ( ) u uu + t u3 = u P 3 E. = 1 [u (t, ) + u (t, ) ] d = const. Natural domain: u(t, ) H 1 (R) Alberto Bressan (Penn State) Nonsmooth evolutions 15 / 36

16 Multi-peakons solutions u(t, ) = n p i (t) e q i (t) i=1 u(0) u(t) q q 1 (t < T) u(t) u( τ) ( τ > T) solutions can lose regularity: u but remain Hölder continuous: u(t, ) H 1 C Alberto Bressan (Penn State) Nonsmooth evolutions 16 / 36

17 Continuous dependence? ( ) u Camassa-Holm : u t + + P = 0 d u(t) v(t) L u(t) v(t) dt FAILS! u(t) v(t) C u(0) v(0) FAILS! Alberto Bressan (Penn State) Nonsmooth evolutions 17 / 36

18 Eample: take v(t, ) = u(t ε, ) u(0) u(t) q q 1 (t < T) u(t) u( τ) ( τ > T) E(t) = [ u (t, ) + u (t, ) ] d constant in time (ecept at t = T ) Alberto Bressan (Penn State) Nonsmooth evolutions 18 / 36

19 As t T, u(t) and u(t ε) become orthogonal in H 1 u(t ε) u(t) H 1 u H 1 u(t ε) u(t) u (t) u (t ε) Alberto Bressan (Penn State) Nonsmooth evolutions 19 / 36

20 Equivalent form of the Camassa-Holm equation (A.B. - Adrian Constantin, ARMA 007) (148 citations!?) Energy variable: ξ R, constant along characteristics t y(t, ξ) y t = u(t, y) y(0, ξ) = ȳ(ξ) ȳ(ξ) 0 (1 + ū ) d = ξ U = u t θ = arctan u T P q = (1 + u ) y ξ Alberto Bressan (Penn State) Nonsmooth evolutions 0 / 36

21 Conservative Solutions u t + (u /) + P = 0 System of ODEs with non-local source terms U = P t θ = (U P) cos θ t 1 θ sin q t = (U 1 ) P sin θ q U(0, ξ) = ū ( ȳ(ξ) ) θ(0, ξ) = arctan ū (ȳ(ξ) ) q(0, ξ) = 1 P(t, ξ) = 1 { ep } ξ θ(s) cos q(s) ds ξ [ ] U (ξ ) cos θ(ξ ) + 1 sin θ(ξ ) q(ξ ) dξ. The global conservative solution can be obtained as the unique fied point of a contractive transformation Alberto Bressan (Penn State) Nonsmooth evolutions 1 / 36

22 Note: The previous construction yields a family of solutions such that ū n ū H 1 0 = u n (t) u(t) L 0 This is not enough to derive error formulas for approimate solutions, or prove uniqueness Alberto Bressan (Penn State) Nonsmooth evolutions / 36

23 A distance functional defined by an Optimal Transport problem for conservative, spatially periodic solutions to Camassa-Holm equation (A.B. and M. Fonte Methods & Applications of Analysis, 005) X = R R T T = R/ZZ Given u H 1 (R), Graph(u) = define { (, u(), arctan u () ) ; R} X Alberto Bressan (Penn State) Nonsmooth evolutions 3 / 36

24 µ u = measure supported on Graph(u), having density 1 + u w.r.t. Lebesgue measure A (, u(), arctan u ()) B a b µ u (A) = b a ( 1 + u () ) d µ u (B) = 0 Alberto Bressan (Penn State) Nonsmooth evolutions 4 / 36

25 A Constrained Optimal Transportation Problem Given u, v H 1, consider the corresponding measures µ u, µ v Let ψ : R R be an absolutely continuous, increasing map. Move the mass µ u to µ v, from to ψ(). Transportation cost: J ψ (u, v) = 1 0 [distance] [transported mass] [ecess mass] v u +d ψ() ψ () + ψ ()d Alberto Bressan (Penn State) Nonsmooth evolutions 5 / 36

26 v u +d ψ() ψ () + ψ ()d total ecess mass: (1 + u ()) (1 + v (ψ()))ψ () d Distance functional: J(u, v) = inf ψ Jψ (u, v) u, v H 1 Key inequality: d dt J( u(t), v(t) ) κ J ( u(t), v(t) ) = uniqueness κ is uniformly bounded on bounded subsets of H 1. Alberto Bressan (Penn State) Nonsmooth evolutions 6 / 36

27 Why a transportation distance is better? u(t) v(t) u(0) v(0) Alberto Bressan (Penn State) Nonsmooth evolutions 7 / 36

28 The Hunter-Saton Equation ( ) u u t + = 1 4 ( ) u d, u(0, ) = ū(). u (t, ) L (square integrable derivative) Possible gradient catastrophe: (u ) t + u(u ) = u Smooth solutions conserve energy: (u ) t + [ u(u ) ] = 0 Alberto Bressan (Penn State) Nonsmooth evolutions 8 / 36

29 Removing a singularity (u ) t + u(u ) = u, θ. = arctan u d dt θ(t, (t)) = θ t + uθ = u 1 + u = cos θ 1 θ = arctan u u = tan θ u 1 + u = tan (θ/) 1 + tan (θ/) = θ sin = 1 cos θ Alberto Bressan (Penn State) Nonsmooth evolutions 9 / 36

30 Odd solutions to Hunter-Saton equation Burgers Hunter Saton u t + (u /) = 1 0 u d Alberto Bressan (Penn State) Nonsmooth evolutions 30 / 36

31 Dissipative solution: Conservative solution: (possibly as a singular measure) 0 u d decreasing in time 0 u d constant in time u 0 u 0 u dissipative u conservative 0 Alberto Bressan (Penn State) Nonsmooth evolutions 31 / 36

32 Relation with the Camassa-Holm Equation ( ) u u t + + P = 0 P = 1 ( ) e u + u ( e ) Leading approimation = + ( ) u u t + = 1 4 ( ) u d + φ φ = 0 = Hunter-Saton equation Alberto Bressan (Penn State) Nonsmooth evolutions 3 / 36

33 A Distance Functional for the Hunter-Saton Equation u v +d ψ() ψ () + ψ ()d ψ : R R smooth, increasing determines a transport plan of the measure u onto the measure v, on the graphs of the two functions (, u()) (ψ(), v(ψ())) d (u, v) = minimum cost among all transportation plans Alberto Bressan (Penn State) Nonsmooth evolutions 33 / 36

34 Transport distance between mass distributions immersed in a fluid Assume: velocity of fluid is Lipschitz continuous with Lipschitz constant ω t=t µ µ µ t=0 µ d (µ t, µ t ) e ωt d (µ 0, µ 0 ) Alberto Bressan (Penn State) Nonsmooth evolutions 34 / 36

35 Back to Hunter-Saton u t + uu = 1 0 u d, (u ) t + [ u(u ) ] = 0 u u(t,) u(t,) 1 The measure u d on the graph of the solution is transported in the -u plane horizontal speed = u vertical speed = 1 0 u d (constant for each particle) Alberto Bressan (Penn State) Nonsmooth evolutions 35 / 36

36 Estimates for the transport distance d (u(t), v(t)) e ωt d (u(0), v(0)) - valid for solutions of Hunter-Saton - similar ideas work for Camassa-Holm Alberto Bressan (Penn State) Nonsmooth evolutions 36 / 36