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The Dirichlet Problem for the Total Variation Flow
, 2001
"... 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 th ..."
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Cited by 20 (7 self)
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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.
On the time continuity of entropy solutions
"... We show that any entropy solution u of a convection diffusion equation ∂tu+divF(u)−∆φ(u) = b in Ω×(0, T) belongs to C([0, T),L 1 loc(Ω)). The proof does not use the uniqueness of the solution. 1 The problem, and main result Convection diffusion equations appear in a large class of problems, and hav ..."
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Cited by 7 (4 self)
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We show that any entropy solution u of a convection diffusion equation ∂tu+divF(u)−∆φ(u) = b in Ω×(0, T) belongs to C([0, T),L 1 loc(Ω)). The proof does not use the uniqueness of the solution. 1 The problem, and main result Convection diffusion equations appear in a large class of problems, and have been widely studied. We consider in the sequel only equations under conservative form: ∂tu + divF(u) − ∆φ(u) = b, (1) so that we can give some sense to (1) in the distributional sense. In this paper, we consider entropy solutions of (1) that do not take into account any boundary condition, or condition for x  → +∞. The proof does not use a L 1contraction principle (see e.g. Alt & Luckaus [1] or Otto [10]), so that it can be applied in case where uniqueness is not insured, like for example complex spatial coupling of different conservation laws as in [3], or for cases where uniqueness fails because of boundary conditions or conditions at x  = +∞, as it will be stressed in the sequel. Let us now state the required assumptions on the data. Let Ω be an open subset of R d (d ≥ 1), and let T be a positive real value or +∞. F is a continuous function, φ is a nondecreasing Lipschitz function, u0 ∈ L 1 loc(Ω). One has to make the following assumption on the source term: (H1) (H2)
Finite volume scheme for twophase flow in heterogeneous porous media involving capillary pressure discontinuities
 M2AN Math. Model. Numer. Anal
"... We study a onedimensional model for twophase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission condi ..."
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Cited by 5 (4 self)
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We study a onedimensional model for twophase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existenceuniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model. MSC subject classification. 35R05, 65M12