(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We 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 →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitz-continuous boundary ∂Ω. We are interested in the problem ∂u Du = div(

During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multi-phase flow in heterogeneous porous media. The purpose of these lecture notes is to review some of this activity. We illustrate the main ideas behind these novel operator splitting algorithms for a basic two-phase flow model. Special focus is posed on the numerical solution algorithms for the saturation equation, which is a convection dominated, degenerate convection-diffusion equation. Both theory and applications are discussed. The general background for the reservoir flow model is reviewed, and the main features of the numerical algorithms are presented. The basic mathematical results supporting the numerical algorithms are also given. In addition, we present some results from the BV solution theory for quasilinear degenerate parabolic equations, which provides the correct ...

We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [31] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase ow equation and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present cont...

We study the backward Euler method with variable time-steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive time-steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character.

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.

We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also studied. It is shown that the proles are non zero solutions of an eigenvalue type problem which seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour enterely dierent to the case of the problem associated to the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties which are peculiar of this special class of quasilinear equations. Key words: Total variation ow, nonlinear parabolic equations, asymptotic behaviour, eigenvalue type problem, propagation of the support. AMS (MOS) subject classication: 35K65, 35K55. 1 Introduction Let be a ...

. We analyse implicit monotone finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion equations in one spatial dimension. Since we allow strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. We thus choose to work with weak solutions that belong to the BV class and, in addition, satisfy an entropy condition. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. This paper complements our previous work [8] on explicit monotone schemes. 1. Degenerate Convection-Diffusion Equations We are interested in finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion problems of the form @ t u + @ x f(u) = @ x (k(u)@ x u); u(x; 0) = u 0 (x); (1) where (x; t) 2 QT = R\Theta (0; T ) and u 0 ; f; k are given, sufficiently smooth functions. For later use, we need a conservative-form version of (1), @ t u + @ x f(u) = @ 2 x K(u)...

. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form (P) @ t u + @xf(u) = @xA (b(u)@xu) ; u(x; 0) = u 0 (x); A(s) = Z s 0 a(¸) d¸; a(s) 0; b(s) 0; under the very general structural condition A(\Sigma1) = \Sigma1. To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV ent...

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [19] and the prescribed mean curvature flow [15]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h, k → 0, and to the total variation gradient flow problem as h, k, ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h²). In particular, it is shown that all error bounds depend on 1/ε only in some lower polynomial order for small ε.

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