Entropy solutions via JKO scheme for a class of degenerate convection-diffusion equations Marco Di Francesco Departament de Matem` atiques Universitat Aut` onoma de Barcelona Joint work with Daniel Matthes (TU Munich) June 28, 2012 Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 1 / 38
Table of contents 1 Preliminaries and main result 2 Contractive flows 3 Flow interchange 4 Construction of entropy solution 5 Uniqueness 6 Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 2 / 38
Preliminaries and main result Table of contents 1 Preliminaries and main result 2 Contractive flows 3 Flow interchange 4 Construction of entropy solution 5 Uniqueness 6 Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 3 / 38
Preliminaries and main result Motivation For some evolutionary PDEs, the following two concept of solutions can be formulated: The notion of Wasserstein gradient flow (De Giorgi, Otto, Ambrosio - Gigli - Savar´ e), which can be introduced every time you deal with ∂ρ � ρ ∇ δ F [ ρ ] � ∂ t = div δρ � F [ ρ ] = ρ log ρ dx gives ρ t = ∆ ρ , � F [ ρ ] = ρ W ∗ ρ dx gives ρ t = div ( ρ ∇ W ∗ ρ ), 1 � ρ m dx gives ρ t = ∆ ρ m , m > 1. F [ ρ ] = m − 1 The notion of Entropy solution (Oleinik, Kruˇ zkov), which classically applies to nonlinear conservation laws ρ t + f ( ρ ) x = 0, nonlinear convection diffusion equations ρ t = g ( ρ ) xx + f ( ρ ) x . Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 4 / 38
Preliminaries and main result Significant examples Nonlocal interaction equations with nonlinear diffusion ∂ t ρ = ∆ ρ m + div ( ρ ∇ G ∗ ρ ) = 0 with m > 1 and G ∈ C 2 and G even. Here, both notions have been used (almost at the same time!) to prove uniqueness of solutions. Scalar conservation laws with symmetric inhomogeneous coefficient ∂ t ρ + ∂ x ( m ( ρ ) ∇ V ( x )) = 0 with m concave, V ∈ C 2 and V even. Here, the interpretation as a gradient flow is still partially open, whereas it is (obviously) clear that you can obtain unique entropy solutions under reasonable assumptions. Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 5 / 38
Preliminaries and main result Our model Nonlinear convection diffusion equation with space dependent coefficient: ∂ t u = ( u m ) yy + ( b ( y ) u m ) y , y ∈ R , t ≥ 0 . (1) Porous medium exponent: m > 1. Assumptions on b : b ∈ L 1 ( R ) ∩ W 1 , ∞ ( R ) (2) Initial condition u 0 ∈ L m ( R ), nonnegative with finite second moment. Important remark Here the gradient flow theory will not give you a unique solution for free, whereas the literature [Karlsen-Risebro 2003] already contains a uniqueness result for entropy solutions. Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 6 / 38
Preliminaries and main result The scaling Problem: How to see the gradient flow structure for (1)? Answer: There exists a function a (a1) a ( x ) ≥ a > 0 for all x ∈ R , (a2) a ∈ W 2 , ∞ ( R ), and a bijective change of coordinates y = T a ( x ) on R such that, for a given weak solution u ( y , t ) to (1) with initial datum u 0 ∈ L 1 ( R ), the scaled function ρ ( x , t ) := T ′ a ( x ) u ( T a ( x ) , t ) satisfies ρ 0 ( x ) := ρ ( x , 0) = u ( x , 0) and the equation ρ [ a ( x ) ρ m − 1 ] x � � ∂ t ρ = x , (3) Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 7 / 38
Preliminaries and main result Intuitive idea behind the scaling Start from (3). Substitute ρ ( x , t ) := T ′ a ( x ) u ( T a ( x ) , t ), to get an equation for u ( y , t ): � a ( x ))) m +1 a ( x ) u m − 1 u y ( m − 1)( T ′ ∂ t u ( y , t ) = � a ( x )) m a ′ ( x ) u m � + ( T ′ x = T − 1 ( y ) . � � y Choose � x 1 � − � m − 1 m +1 T a ( x ) = a ( ξ ) d ξ. (4) m 0 to make the diffusion coefficient homogeneous. We obtain (1). Notation: u ( y , t ) = S a [ ρ ( · , t )]( y ). Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 8 / 38
Preliminaries and main result Scaling: from b to a The rigorous definition of the new coefficient a ( x ) for fixed b ∈ L 1 ( R ) ∩ W 1 , ∞ ( R ). Set � y � − ( m − 1) � α ( y ) = α 0 exp b ( η ) d η . 2 m 0 Notice that α ∈ W 2 , ∞ ( R ) and α ≥ α > 0. Consider the unique solution to the Cauchy problem � T ′ ( x ) = α ( T ( x )) , x ∈ R (5) T (0) = 0 , Finally, define a as follows m m − 1 ( α ( T ( x ))) − ( m +1) . a ( x ) := (6) a satisfies: a ∈ W 2 , ∞ ( R ) and a ( x ) ≥ a > 0. Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 9 / 38
Preliminaries and main result To fix ideas: Throughout this talk we shall always think of ρ and u being 1 : 1-related via the scaling u ( y , t ) = S a [ ρ ( · , t )]( y ) . ρ ( x , t ) should be thought as a solution to (3) ρ [ a ( x ) ρ m − 1 ] x � � ∂ t ρ = x , for which we can use a gradient flow (JKO) approach, u should be thought as a solution to (1) ∂ t u = ( u m ) yy + ( b ( y ) u m ) y , y ∈ R , t ≥ 0 , for which we can use the entropy solution approach. Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 10 / 38
Preliminaries and main result Entropy solutions for (1) Following [Carrillo M. - ARMA 1999] and [Karlsen, Risebro - DCDS 2003]: Definition (Definition of entropy solution) Let u 0 ∈ L 1 ∩ L ∞ ( R ), and let T > 0. A non-negative measurable function u :]0 , T [ × R → R is an entropy solution to (1) with initial condition u 0 if u ∈ L 1 ∩ L ∞ (]0 , T [ × R ), if u m ∈ L 2 (]0 , T [; H 1 ( R )), if u ( t ) → u 0 in L 1 ( R ) as t ↓ 0, and if inequality � T � | u − k | ϕ t d y d t 0 R � T � Sgn( u m − k m ) �� ( u m ) y + b ( u m − k m ) � ϕ y − b y k m ϕ � ≥ d y d t . 0 R is satisfied for all nonnegative test functions ϕ ∈ C ∞ c (]0 , T [ × R ), and for all k ∈ R + . Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 11 / 38
Preliminaries and main result Construction of entropy solutions In the literature there are many methods used to construct entropy solutions. Some of them do not apply to our model. Here is a (possibly incomplete) list: Vanishing viscosity approach Wave front tracking algorithm Glimm’s scheme Semigroup approach (implicit Euler) We shall add another method to the list, namely: JKO - De Giorgi scheme for ρ ⇒ scaling u = S a [ ρ ] Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 12 / 38
Preliminaries and main result The JKO scheme for (3) F [ ρ ] := 1 � a ( x ) ρ m d x . m R For a time step τ > 0, F τ ( ρ ; σ ) = 1 2 τ W 2 ( ρ, σ ) 2 + F ( ρ ) , where W 2 ( ρ, σ ) is the 2-Wasserstein distance. 1 ρ 0 τ := ρ 0 . 2 For n ≥ 1, let ρ n τ ∈ P 2 ( R ) be the (unique global) minimizer of F τ ( · ; ρ n − 1 ). τ Piecewise constant interpolation: ρ τ ( t ) = ρ n ¯ for ( n − 1) τ < t ≤ n τ. τ Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 13 / 38
Preliminaries and main result Main result Theorem (DF, Matthes - (almost preprint)) Let ρ 0 ∈ P 2 ( R ) ∩ L ∞ ( R ) , and let a as above. Then every vanishing sequence ( τ k ) k ∈ N of time steps contains a subsequence (not relabeled) such that the ¯ ρ τ k converge to a curve ρ ∗ : [0 , ∞ [ → P 2 ( R ) — in L m (]0 , T [ × R ) , and also uniformly in W 2 on each time interval [0 , T ] — that is continuous with respect to W 2 . The rescaled function u ∗ = S a [ ρ ∗ ] is an entropy solution to (1) in the sense of Definition 1, with initial condition u 0 = S a [ ρ 0 ] . Corollary Since the entropy solution u ∗ = S a [ ρ ∗ ] to (1) is unique, then the JKO scheme for F has a unique limit point. Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 14 / 38
Preliminaries and main result Remarks A similar connection has been pointed out for instance by [Gigli-Otto 2010] in the context of the inviscid Burger’s equation. The interpretation of the coincidence between entropy solutions and gradient flows is that both types of solutions can be characterized by diminishing an underlying entropy functional “as fast as possible”. Apart from revealing an interesting connection between two different theories for nonlinear evolution equations, our simple example indicates a possible general strategy to prove existence and uniqueness of certain degenerate parabolic equations. First, identify a variational structure behind the equation and use methods from the calculus of variations to construct a candidate for a solution; a priori, there might be several. Second, show that this candidate is an entropy solution by deriving further a priori estimates in the variational framework. Third, conclude uniqueness of the entropy solution. Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 15 / 38
Contractive flows Table of contents 1 Preliminaries and main result 2 Contractive flows 3 Flow interchange 4 Construction of entropy solution 5 Uniqueness 6 Open problems and remarks Marco Di Francesco (UAB) Hyp2012, Padova, 25-29 June 2012 June 28, 2012 16 / 38
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