p-Parabolic approximation of total variation flow solutions (original) (raw)

A variational method for a class of parabolic PDES

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2011

In this manuscript we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming the initial condition to be a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Then De Giorgi's interpolation method reveals to be a powerful tool for proving convergence of our algorithm. Finally we show uniqueness and stability in L 1 of our solutions.

Variational parabolic capacity

Discrete and Continuous Dynamical Systems, 2015

We establish a variational parabolic capacity in a context of degenerate parabolic equations of p-Laplace type, and show that this capacity is equivalent to the standard capacity. As an application, we compute capacities of several explicit sets.

Minimizing total variation flow

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

We prove existence and uniqueness of weak solutions for the minimizing Total Variation ow with initial data in L 1 . W e prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t ! 1 . We also prove that local maxima strictly decrease with time, in particular, at zones immediately decrease their level. We display some numerical experiments illustrating these facts.

A Variational Inequality for Discontinuous Solutions of Degenerate Parabolic Equations

2000

The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution

On a variational approach to the Navier-Stokes Equations

2020

 (i) ∂v ∂t + divx (v ⊗ v) +∇xp = ν∆xv + f ∀(x, t) ∈ Ω× (0, T ) , (ii) divx v = 0 ∀(x, t) ∈ Ω× (0, T ) , (iii) v = 0 ∀(x, t) ∈ ∂Ω× (0, T ) , (iv) v(x, 0) = v0(x) ∀x ∈ Ω . Here v = v(x, t) : Ω×(0, T ) → R is an unknown velocity, p = p(x, t) : Ω×(0, T ) → R is an unknown pressure, associated with v, ν > 0 is a given constant viscosity, f : Ω× (0, T ) → R is a given force field and v0 : Ω → R N is a given initial velocity. The existence of weak solution to (1.1) satisfying the Energy inequality was first proved in the celebrating works of Leray (1934). There are many different procedures for constructing weak solutions (see Leray [9],[10] (1934); Kiselev and Ladyzhenskaya [8] (1957); Shinbrot [12] (1973)). The most common methods are based on the so called Faedo-Galerkin approximation process. Application of Faedo-Galerkin method for (1.1) was first considered by Hopf in [7]. We also refer to Masuda [11] for the problem in higher dimension. In this paper we present a variational met...

A variational approach to the Navier–Stokes equations

Bulletin des Sciences Mathématiques, 2012

We propose a time discretization of the Navier-Stokes equations inspired by the theory of gradient flows. This discretization produces Leray/Hopf solutions in any dimension and suitable solutions in dimension 3. We also show that in dimension 3 and for initial datum in H 1 , the scheme converges to strong solutions in some interval [0, T) and, if the datum satisfies the classical smallness condition, it produces the smooth solution in [0, ∞).