Singular Diffusion Equations With Nonuniform Driving Force. Y. Giga University of Tokyo (Joint work with M.-H. Giga) July 2009

Transcription

1 Singular Diffusion Equations With Nonuniform Driving Force Y. Giga University of Tokyo (Joint work with M.-H. Giga) July

2 Contents 0. Introduction 1. Typical Problems 2. Variational Characterization 3. Definition of Solutions 4. Comparison Principle Idea of the Proof 2

3 0. Introduction curvature flow with a singular interfacial energy on Γ 2 t R V = M( n) ( κγ + σ) γ = γθ ( ): n = (cos θ,sin θ ) : interfacial energy density normal of Γ t n κ = ( γ ( θ) + γ( θ)) κ : γ σ : driving force, M ( n ) > 0: γ + γ =Σc δ( θ a ) k k weighted curvature mobility crystalline energy: typical singular energy! Γ t 3

4 1. Typical Problems: ( σ :given) ( a ) u = (sgn u ) + σ ( x), x R, t > 0 t x x with I.C. ux (,0) = u 0 More generally, ( b ) t x x x u = a( u )[( W ( u )) +σ ( x)] W : convex, a > 0 ( W( p) = p, a 1) a const Non divergence form 4

5 Feature Energy density W has jump discontinuities so that diffusion is singular. ( a ) is of the form u δ u u σ x = 2 ( ) + ( ). t x xx 5

6 Source of Problems (ⅰ) Total variation flow: (a) (ⅱ) Crystalline curvature flow with (b) driving force term: W( p) : piecewise linear, convex 2 1/2 a( p) = (1 + p ), σ = constant Angenent-Gurtin 89, Taylor 91 6

7 Further Applications (Vertical Diffusion) Burgers eq. u + uu = 0 V t = y on Γ (*) t x Γ = {( xy, ); y= uxt (, )} (graph of u) t A solution of (*) may overturn and cannot be viewed as the graph of entropy solution. Consider V = y M div pγ( n), γ( p1, p2) = p2 instead of (*), where M > 0. 7

8 Thm (M.-H. Giga Y. G. 03) If M > 0 is sufficiently large (with respect to jump size), then overturn is prevented and Γ t becomes the graph of entropy solution at least for Riemann problem. cf. Y. G. 02, Y.-H. R. Tsai Y.G. Osher 02, Y. Brenier: 09 formulation by an obstacle function 8

9 Notion of Solutions (1) Subdifferential formulation: (a) (not (b)) u Goal: Extend the theory for depending on! t E( u) σ : general Fukui-Y.G. 96 (2) Viscosity approach: (b) a = const Mi-Ho Giga-Y.G. 98, 99, 01 σ : const, u + F( u, ( W ( u )) + σ ) = 0 x t x x x σ 9

10 Nonlocal Quantity Consider simplest eq ( W( p) = p, σ 0) u = (sgn u ). t x x What is the speed of the facet (flat part)? a b x Assume facet stays as facet b+ δ a δ Crystalline flow u = W ( δ ) W ( + δ ) t 2 ut = ( δ 0) b a δ = u ( a δ), δ = u ( b+ δ) for admissible polygon. x x 10

11 What is the quality Non uniform σ σ W x x Λ ( u) ( x) = ( W ( u )) +σ ( x)? Λ σ W In general is not constant on the facet so that facet may break. 11

12 Speed of Evolution (a) Thm (Komura, Brezis - Pazy) H : Hilbert space, E : convex, lower semicontinuous, There exists unique solution solving Moreover, u is right differentiable for all t > 0 and canonical restriction / minimal section: [ ) [ δ ] u C(0,, H) AC(, T, H) du E( u) a.e. t 0, u(0) u0. dt > = + d u dt = 0 = H E( u) argmin{ f ; f E( u)} 0 Eu ( ). u0 H Solution knows how to evolve! 12

13 How to calculate the Speed Eu Wu xu x H 2 ( ) = { ( x ) σ( ) } d, T= R/ Z, = L ( T), T f E( u) f = η σ, η( x) W( u ( x)), f = 0 0 E( u) x 0 2 f f x f T = argmin{ d ; = η σ, x x η( x) W( u ( x)) }. x Obstacle type condition at the place where the slope belongs to the jump of W. 13

14 Explicit Solutions (1) Angenent-Gurtin, Taylor (2) σ : const σ σ σ admissible polygon ( x) = ( x), > 0, e.g. a( p) = (1 + p ) x Explicit solution starting with u. (bending solution) /2 Nonlocal Hamilton-Jacobi equations with unusual free boundary Y.G.-Gorka-Rybka (2010?) Y.G.-Rybka (2009) [M.-H. Giga-Y.G. 98 for (a)] 14

15 2. Variational Characterization Λ σ W From (a) we learn the speed is given by solving an obstacle problem. I = ( a, b), W( p) = p for simplicity. Admissible set K I η χχ H I η η a χ η b χ 1 () = { (): 1, () =, () } + = +, ( χ χ takes either or + 1 )

16 Energy and Curvature J ( w, I ) = w + σ χ χ dx + I w χ χ+ = arg min{ J ( w, I) w= η, η K ( I)} 0 χ χ x χ χ u :faceted with slope zero on with transition number Define χ χ + σ ( ) ( ): ( ) ( ), W u x w χ χ + 0 x σ x x I Λ = + I 16

17 Transition number of near facet u I χ = 1 χ + = 1 a graph of u I b χ, χ + takes either or + χ = +1 χ + = 1 χ : left transition number a I b χ + : right transition number 1 if ux ( ) ua ( ) for x anear a χ = 1 if ux ( ) ua ( ) for x anear a 17

18 Example (1) σ W u x w 0 x σ x Λ ( ) ( ) = ( ) + ( ) graph of u I (2) σ W u x w + 0 x σ x Λ ( ) ( ) = ( ) + ( ) + graph of u I 18

19 More on Values of The case σ const (1) (2) Λ σ W +1 graph of η a b σ 2 Λ W = + σ b a 1 a Λ = 0 +σ σ W b 19

20 Nonconstant Driving Force: Reformulation Better to rewrite problems: Formally set ξ x = w+ σ Kχ χ I ξ H I Z ξ Z + 1 ( ) = { ( ) : (obstract condition) ξ( a) = Z( a) χ, ξ( b) = Z( b) + χ ( BC..)} + Here x Z( x) = σ dx: primitive of σ. { } 2 χχ χχ + w0 + σ : = argmin ξ dx ξ K I + 20

21 Obstacle Problem optimal ξ Ex. σ ( x) = 2 x, I = ( a, a) χ = 1, χ = 1 + a: not small a 0 a graph of u ξ = const. on I I is calibrable (Cheeger set) I speed profile ( = ξ ) 21

22 Comparison of curvatures Assume: σ is locally Lipschitz on R. Lemma u u 1 2, u : faceted on I ( i = 1, 2) i i u = u on I I Λ σ W Λ σ 1 W 2 ( u ) ( x) ( u ) ( x) for x I I 1 2 u 1 u 2 22

23 Stability Lemma χ χ 0 + Λ ( χ, χ, I, x) = w ( x) + σ( x), x I + For each r > 0 up{ Λ( χ, χ, I, x) Λ( χ, χ, I μ, x μ) : + + I < r, x I} 0 as μ 0. 23

24 3. Definition of Solutions We consider energy density W s.t. (ⅰ) (ⅱ) (ⅲ) W :convex P finite set for any compact set in ((ⅳ) jumps on P ) W C 2 ( R\ P) sup { W ( p) : K ( R\ P)} < W K R. 24

25 W Ex. : piecewise linear convex (crystalline) What is a suitable class of test functions to define viscosity like solution? σ = const (c.f. M.-H. Giga Y.G. 98 ) 25

26 Faceted function f C 2 ( Ω), Ω: interval We say that is -faceted if for with I P x Ω 0 0 closed interval s.t. f f ( x ) P (ⅰ) (ⅱ) (ⅲ) x I Ω 0 f ( x) P f ( x) P for for x I x nbd \ I (ⅳ) transition number is defined. 26

27 graph of f I I I : faceted region 27

28 Admissible functions 2 2 CP ( Ω ) = { f C ( Ω ) f : P faceted} ϕ C 1,2 ( Q), Q =Ω (0, T) ϕ ϕ is admissible if is of the form ϕ ( x, t) = f( x) + g( t) 2 1 P f C ( Ω), g C (0, T). AP ( Q ) : the set of admissible functions 28

29 Definition (Global version) t x x x u + F( u,( W ( u )) + σ ( x)) = 0 u C( Q) F σ t x W ϕ ϕ ϕ whenever Q : subsolution in if + (, Λ ( )) 0 at xˆ tˆ (, ) max( u ϕ) = ( u ϕ) ( xˆ, tˆ ) for ϕ AP ( Q ). W W x x σ ϕ ϕ σ ˆx ( Λ ( ) = ( ( )) + if faceted region) 29

30 Jet in faceted region J + P ( xt ˆ, ˆ) = { R : ϕ τ ω modulus ( xt, ) ( xt ˆ, ˆ) + p ( x xˆ) + ( t tˆ) ϕ ϕ τ + ω( t tˆ ) t tˆ in N ( tˆ δ, tˆ+ δ)}, N where is a semi nbd of faceted region. Here ϕ AP ( Q ) 30

31 Set N [ ] [ ] I N I N [ ] [ ] N I I N 31

32 Admissible superfunction ϕ USC( Q), ϕ(, tˆ ) C( Ω) ϕ (A) is admissible superfunction at ϕ J if one of following hold. (, tˆ ) : + p ϕ P ( xˆ, tˆ ) -faceted at and : non empty ˆx ( xt ˆ, ˆ) 32

33 (B) (C) + (, τ p, X) P ϕ(,) xˆ tˆ ϕ(, x): P -faceted at and is on bdry of the faceted region + (, τ p,0) P ϕ(,) xˆ tˆ ˆx ˆx 33

34 u if u C( Q) ϕ at Equivalent Definition (infinitesimal version) subsolution in : admissible superfunction testing ( xt ˆ, ˆ) Q from above satisfies (ⅰ) for all + F( ( xˆ, tˆ), Λ ( )( xˆ, tˆ)) 0 x σ W τ ϕ ϕ τ J + P ϕ ( xt ˆ, ˆ) if (A) holds; 34

35 (ⅱ) (ⅲ) τ + F( p, W ( p) X + σ ( xˆ )) 0 (, τ px, ) P ϕ(,) xt ˆ ˆ + F( p, σ ( xˆ )) 0 (, τ p,0) P + ϕ(,) xˆ tˆ τ ˆx x for all for all if (B) holds; if (C) holds and is a strict max of near in direction. + u ϕ ˆx (cf. M.-H-.Giga-Y.G. 2001) 35

36 4. Comparison Principle u + F( u,( W ( u )) + σ ( x)) = 0 t x x x Assumption (F1) (F2) (F3) F C( R R) for F( p, X) F( p, Y) X Y F( px, ) FpY (, ) C(1 + p ) X Y (Lipschitz continuity) 36

37 Example t x x x u a( u )[ W ( u )) + σ ( x)] = 0 a :continuous and nonnegative Assumption (ⅲ) is standard if we consider the case that ; eq is u + H( u, x) = 0. t x x W 0 ( -depending HJ equations) 37

38 Thm Comparison Principle Assume that satisfies (F1)-(F3). σ Assume that is Lipschitz on where Ω is a bdd interval. u F v : sub : super Ω u * v * on * p Q u v in Q * 38

39 Idea: u, v USC( Q) Φ ( x, yts,, ): = uxt (, ) vys (, ) α( t s) 1 γ γ B(( x y) β ) β T t T s α, β Argue by contradiction. γ : large : small 2 B 2 P C B (0) = 0 : convex function, B( x) 2 x for large x, B( x) 0 graph of B 39

40 γ α : small, : large fixed. maximizer ( x, y, t, s ) β β β β May assume that are not on the bdry of faceted region. It is in a interior of faceted region for large. x β, y β β 40

41 Max Principle and (faceted) sup volution U 1 U 2 admissible superfunction admissible subfunction s.t. lengths of faceted region agree 41

42 Sup Convolution with Faceted Functions M.-H. Giga Y. G ( x ρ) ρ, x> ρ, ϑ( x, ρ) = 0, x ρ, x + x< 2 ( ρ) ρ, ρ, α u ( x) = sup { u( ξ) ϑ( x ξ, α)}. ξ u 0 If attains a local maximum at, then is 0 slope faceted at. u α x x 0 42

43 y β x β Use definition of infinitesimal version γ 2 α σ ( tβ sβ) + + F( p, Λ ( 2 W U1)( xβ, tβ)) 0 T γ σ 2 α( tβ sβ) + F( p, Λ ( 2 W U2 )( yβ, sβ)) 0 T (cf. M.-H. Giga-Y.G. 2001) (1) (2) 43

44 σ W σ 1 β β W 2 β β Λ ( U )( x, t ) Λ ( U )( y, s ) o(1) as x y 0 ( ). β β β Proved by Comparison of curvature and Stability 44

45 Recall P : finite set F( p, X) F( p, Y) 0 as X Y 0 F( px, ) F( py, ) if X Y Then (1)-(2) yields as 2γ 2 T 0 β x y 0. so that β β Contradiction! 45

46 Related Problems 3-dimensional problem is widely open. only studied when σ 0. Even in this case facet may break. Bellettini, Paolini, Novaga 46

47 Further possible development Problems Stability: approximation smooth energy σ ( :const; M.-H. Giga Y. Giga 1999) Level set approach: σ [ : const; M.-H. Giga Y. Giga 2001] Convergence of Allen-Cahn type σ = 0 [, Ohtsuka Schätzle Y. G. 2006] [Bellettini Goglione Novaga 2000] (Convergence to crystalline flow) 47

48 Special Project A minisemester on evolution of interfaces, Sapporo 2010 July 12, August 13, 2010 Organizers: Y. Giga (Tokyo), H. Ishii (Tokyo), T. Funaki (Tokyo), Y. Tonegawa (Sapporo), R. V. Kohn (New York), P. Rybka (Warsaw) Venue: Department of Mathematics, Hokkaido University Intensive Activities (1) Tutorial Lectures and Interdisciplinary Conference Mathematical Aspects of Crystal Growth In cooperation with SIAM July 26 -July 30, 2010 Organizers: Y. Giga (Tokyo), H. Ishii (Tokyo), R. Kohn (New York), P. Rybka (Warsaw), E. Yokoyama (Tokyo) Tutorial Lecturers: R. Caflisch (Los Angeles), D. Margetis (College Park), R. Monneau (Paris) (2) Tutorial Lectures and International Workshop Singular Diffusion and Evolving Interfaces August 2 -August 6, 2010 Organizers: T. Funaki (Tokyo), Y. Giga (Tokyo), P. Rybka (Warsaw), Y. Tonegawa (Sapporo) Tutorial Lecturers: G. Bellettini (Rome), B. Kawohl (Cologne) Sponsored by Grant-in-Aid for Scientific Research (S) ( ), JSPS Supported by Research Center for Integrative Mathematics, Hokkaido University. 48