The method of viscosity solutions for analysis of singular diffusion - PowerPoint PPT Presentation

Mathematics for Nonlinear Phenomena: Analysis and Computation Sapporo, August 2015 The method of viscosity solutions for analysis of singular diffusion problems appearing in crystal growth problems Piotr Rybka Joint project with Y.Giga, M.-H.Giga, P.Górka, M.Matusik, P.B.Mucha Content: 1. Gibbs-Thomson relation – the weighted mean curvature flow 2. Viscosity solution for ( ) u t = a ( u x ) W p ( u x ) x + σ special cases of W : W ( p ) = | p | , W ( p ) = | p + 1 | + | p − 1 | . 3. Advantages and disadvantages of viscosity solution.

1. Setting up the geometric problem We study equations behind the Gibbs-Thomson relation, i.e. βV = κ γ + σ on Γ( t ) . (1) R 2 – a curve, n its outer normal, β = β ( n ) – a kinetic coefficient. a) Γ( t ) ⊂ I b) Formally, the weighted mean curvature is ( ) κ γ = − div S ∇ ζ γ ( ζ ) | ζ = n ( x ) . (2) If γ ( ζ ) = | ζ | , then κ γ is the Euclidean mean curvature. Here, γ ( p 1 , p 2 ) = | p 1 | γ Λ + | p 2 | γ R , (3) c) σ – the driving (supersaturation, temperature, pressure, ...). It should be a solution to ϵσ t = ∆ σ, ϵ = 0 or ϵ = 1 . (4) Full scale numerical simulations for (1), (4) with smoothed out γ and β were perfomed by Garcke and co-workers (2013), but there is no corresponding theory. Here, σ is given. 1

We will study (1) for graphs or special closed curves, which are small Γ γ ( n ) d H 1 under ∫ perturbations of the shape minimizing the surface energy the volume constraint. The minimizer is a scaled Wulff shape, W γ , i.e. a R 2 , γ ) . For γ given by (3) the Wulff shape is a ball in the space dual to (I rectangle. Our closed curves of special interest are small perturbations of W γ , bent rectangles , R S+ x x =d (t,x ) 2 R L 2 1 1 Λ L 0 x =d (t,x ) 1 1 2 l 1 l 0 R 0 x −r −r r r R 1 1 0 0 1 1 S+ Λ Fig.: A bent rectangle 2

When we talk about a bent recangle, we assume that each side of a bent rectangle is a graph of a Lipschitz function such that it has three facets. Basically, a connected part of Γ( t ) , F , will be called a facet , if: 1) the normal to F is one of singular normals i.e. ( ± 1 , 0) , (0 , ± 1) , the nor- mals to the Wullf shape W γ ; 2) F is maximal with respect to set inclusion having property 1). We study (1) for graphs of u or bent rectangles when γ given by (3). 3

Summary of know results 1) Novaga, Chambolle using variational tools constructed the flat flow of R n . Restriction: convex body as a datum. V = κ γ in I 2) Andreu, Caselles, Mazon, Moll used nonlinear semigroup methods to    ∇ u R n . Anisotropic counterparts studied by Moll too.  in I solve u t = div |∇ u | R 2 3) Y.Giga, P .Górka, PR used variational tools to study βV = κ γ + σ in I for special closed curves and special σ . ( ) 4) Y.Giga, M.-H.Giga developed viscosity theory for u t = M ( u x ) W p ( u x ) x + σ . 5) P .Mucha, PR proved existence of solutions to u t = ( sgn u x ) x + ϵu xx , ( ϵ = 0 or ϵ = 1 ) by smooth approximation. Today, I would like to present an apllication of approach 4) to 3) and 5). 4

Our initial approach to study bent rectangles as variational solutions to (1) was the following: the corner is defined as the intersection of two facets of evolving graphs. Thus, it is sufficient to look at two graphs d R ( t, · ) , d Λ ( t, · ) . √ Set u = d R ( t, x 1 ) , then n = ( − u x 1 , 1) / u 2 x 1 + 1 . Hence, ∇ γ is not well defined on facets! Remark. If we stick u and n into (1), then we obtain     u x     β ( u x ) u t =  γ Λ sgn + σ         √ u 2     x 1 + 1    x or equivalently ( ) u t = M ( u x ) W p ( u x ) x + σ , (5) where W ( p ) = γ Λ | p | . 5

Convexity of γ permits us to use the subdifferential, ∂γ , (defined for all R 2 ) in place of ∇ ζ γ ( ζ ) | ζ = n ( x ) in (2). We recall that the subdifferential xi ∈ I is the set of all supporting hyperplanes, e.g. the subdifferential of f ( x ) = | x | at x = 0 is depicted below. Fig.: subdifferential ∂f (0) . 6

The use of ∂γ requires finding ξ , a selection ∂γ . Thus, the operator σ − div S ∇ ζ γ ( ζ ) | ζ = n ( x ) (6) reduces to σ − ∂ξ ∂x, where ξ ∈ ∂γ . Our construction of variational solutions is based on the right selection of ξ . It is based of the observation that (6) is the E-L of functionals S Λ | σ − div S ξ | 2 H 1 , S R | σ − div S ξ | 2 H 1 . ∫ ∫ E Λ ( ξ ) = E R ( ξ ) = (7) We will show that variational solutions are viscosity solutions , in the sense developed by M.-H.Giga and Y.Giga. Here come the bonuses: uniqueness of variational solutions as well as preservation of verteces. 7

2. Variational solutions A family of couples (Γ( t ) , ξ ( t )) t � 0 will be called a variational solution iff Γ( t ) is a bent rectangle and ξ ( t ) is a solution to min {E Λ ( ξ ) : div S ξ ∈ L 2 , ξ ( x ) ∈ ∂γ ( n ( x )) } , min {E R ( ξ ) : div S ξ ∈ L 2 , ξ ( x ) ∈ ∂γ ( n ( x )) } and eq. (1) is satisfied in the L 2 sense. An advantage of variational solutions is that they are ‘explicit’ compared to viscosity solutions. However, at a certain level of complication of the geometry of the data this advantage is lost. 8

Proposition 2.1 Taking into account the form of the minimizers of E R , E Λ , then (1) on the sides S R and S Λ becomes − σ ( t, R 0 , s ) ds + γ R ∫ l 0 ˙ R 0 /M (0) = on [0 , l 0 ] 0 l 0 d Λ t = σ ( t, d Λ , s ) M ( d Λ x ) for s ∈ ( l 0 , l 1 ) ∫ L 1 2 γ R ˙ R 1 /M (0) = − σ ( t, R 1 , s ) ds + on [ l 1 , L 1 ] l 1 L 1 − l 1 − σ ( t, s, L 0 ) ds + γ Λ ∫ r 0 ˙ L 0 /M (0) = on [0 , r 0 ] (8) 0 r 0 d R t = σ ( t, s, d R ) M ( d R x ) for s ∈ ( r 0 , r 1 ) ∫ R 1 2 γ Λ ˙ L 1 /M (0) = − σ ( t, s, L 1 ) ds + on [ r 1 , R 1 ] . r 1 R 1 − r 1 Here, M ( d x ) = 1 /β ( d x ) . Note. System (8) is not closed until we specify evolution of r 0 ( · ) , r 1 ( · ) , l 0 ( · ) , l 1 ( · ) , these are genuine free boundaries. (8) is a system of Hamilton-Jacobi eqs with discontinuous Hamiltonians. 9

Theorem 2.2 (Giga, Górka, PR 2013) Let us suppose that Γ(0) is a bent rectangle such that d Λ 0 ∈ C 2 ([ l 0 , l 1 ]) , d R 0 ∈ C 2 ([ r 0 , r 1 ]) . Moreover, σ satisfies σ ( x 1 , x 2 ) = σ ( ± x 1 , ± x 2 ) and x i ∂σ ∂x i ( x 1 , x 2 ) > 0 , x i ̸ = 0 , i = 1 , 2 . ( Berg’s effect ). We assume that one of the following con- ditions occurs at each interfacial point r i , l i , i = 0 , 1 . r 0 < 0 (resp. ˙ (a) condition (9) holds at r 0 (resp. l 0 ) and ˙ l 0 < 0 ), i.e. the facet shrinks ∫ r 0 − σ ( t, s, L 0 ) ds + γ Λ at r 0 : σ ( t, r 0 , L 0 ) = , (9) 0 r 0 r 0 > 0 (resp. ˙ l 0 > 0 ) and d R x ( r 0 (0)) > 0 (resp. d Λ or (b) ˙ x ( l 0 (0)) > 0 ), i.e. the facet expands. (Similar conditions at r 1 and l 1 ). Then, under some additional technical restrictions on data, there exists a variational solution to (1). If all the interfacial curves satisfy (9), (10), ∫ R 1 2 γ Λ at r 1 : σ ( t, r 1 , L 1 ) = − σ ( t, s, L 1 ) ds − . (10) r 1 R 1 − r 1 then the solution is unique. 10

Comments on (9) and (10). Characteristics of the Hamilton-Jacobi eq. d t = σ ( t, d, x ) M ( d x ) interact with the free boundaries r 0 , r 1 , l 0 , l 1 . Any of these curves may be r 0 classical nonlocal ODE Hamilton−Jacobi eq. r x Fig.: a shock wave r 0 00 or classical Hamilton−Jacobi eq. r 0 nonlocal ODE Fig.: a rarefaction wave r 0 r x 00 11

Further comments We know the structure of solutions, but: 1) We are not able to establish existence for all configurations; 2) Uniqueness limited to smooth data; 3) Corners motion is defined in an artificial way, it does not follow from equations. In order to resolve issues 2) and 3) we resort to viscosity theory. 4) Berg’s effect, x i ∂σ ∂x i ( x 1 , x 2 ) > 0 , x i ̸ = 0 , i = 1 , 2 , has been established experimentally, (Berg 1938) and it well-known in the Physics community (Yokoyama, Kuroda 1990, Yokoyama, Sekerka, Furukawa 2000, Nelson 2001). Despite efforts to prove it, (Seeger 1953, Giga, PR 2003), it se- ems that it is a rather rare mathematical phenomenon, related to regularity of solutions (Kubica, PR 2014). 12

On Berg’s effect We studied the following equation,  ∆ u = 0 in Ω := R 2 \ R 1 ,       u = 0 on ∂R 2 ,  (11)  ∂u   ∂ n = u n on ∂R 1 ,     where R 1 = ( − r 1 , r 1 ) × ( − r 2 , r 2 ) , R 2 = λ 0 R 1 , λ 0 > 1 , n is the outer normal to Ω and  a for | x 2 | = r 2 ,   u n =  b for | x 1 | = r 1 .    Theorem 2.3 (A.Kubica, PR 2014) Let us suppose that R 1 is as above. There are unique numbers α, β related with Ω such that | α | + | β | > 0 and if u is a weak solution to (11) then u ∈ C 1 (Ω) ⇐ ⇒ aα + bβ = 0 . Tools used in the proof: the standard representation of singular solutions and careful analysis of the level sets of harmonic functions. 13