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Discrete duality finite volume schemes for LerayLions type elliptic problems on general 2D meshes
 Num. Meth. PDE
"... Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume s ..."
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Cited by 26 (8 self)
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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, MintyBrowder type arguments, and “hyperbolic ” L ∞ weak ⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna,...). Our results cover the case of nonLipschitz nonlinearities.
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 21 (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.
Solute transport in porous media with equilibrium and nonequilibrium multiplesite adsorption: Uniqueness of weak solutions
"... this paper we prove an L ..."
A Note On The Uniqueness Of Entropy Solutions Of Nonlinear Degenerate Parabolic Equations
 J. Math. Anal. Appl
, 2001
"... . Following the lead of Carrillo [6], recently several authors have used Kruzkov's device of \doubling the variables" to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order dierential operator is not allowed to depend exp ..."
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Cited by 11 (5 self)
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. Following the lead of Carrillo [6], recently several authors have used Kruzkov's device of \doubling the variables" to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order dierential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result found in Karlsen and Risebro [14] to a class of degenerate parabolic equations with spatially dependent second order dierential operator. The class is large enough to encompass several interesting nonlinear partial dierential equations coming from the theory of porous media ow and the phenomenological theory of sedimentationconsolidation processes. 1.
Existence and Uniqueness of Solution for a Parabolic Quasilinear Problem for Linear Growth Functionals with L¹ Data
, 2001
"... We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk ..."
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Cited by 2 (0 self)
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We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the timedependent minimal surface equation. We also study the asimptotic behavoiur of the solutions.
An Error Estimate For Viscous Approximate Solutions Of Degenerate Parabolic Equations
 J. Nonlinear Math. Phys
"... . Relying on recent advances in the theory of entropy solutions for nonlinear (strongly) degenerate parabolic equations, we present a direct proof of an L 1 error estimate for viscous approximate solutions of the initial value problem for @ t w + div V (x)f(w) = A(w), where V = V (x) is a vect ..."
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Cited by 1 (0 self)
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. Relying on recent advances in the theory of entropy solutions for nonlinear (strongly) degenerate parabolic equations, we present a direct proof of an L 1 error estimate for viscous approximate solutions of the initial value problem for @ t w + div V (x)f(w) = A(w), where V = V (x) is a vector eld, f = f(u) is a scalar function, and A 0 () 0. The viscous approximate solutions are weak solutions of the initial value problem for the uniformly parabolic equation @ t w " + div V (x)f(w " ) = A(w " ) + "w " , " > 0. The error estimate is of order p ". 1. Introduction In this paper, we are interested in certain \viscous" approximations of entropy solutions of the initial value problem (1.1) ( @ t w + div V (x)f(w) = A(w); (x; t) 2 Q T ; w(x; 0) = w 0 (x); x 2 R d ; where Q T = R d (0; T ) with T > xed, u : Q T ! R is the sough function, V : R d ! R is a (not necessarily divergence free) velocity eld, f : R ! R is the convective ux funct...
L¹Contraction and Uniqueness for Quasilinear EllipticParabolic Equations
"... We prove L¹contraction principle and uniqueness of solutions for quasilinear ellipticparabolic equations of the form @ t [b(u)] \Gamma div[a(ru; b(u))] + f(b(u)) = 0 in (0; T )\Theta\Omega ; where b is monotone nondecreasing and continuous. We only assume that u is a weak solution of finite e ..."
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We prove L¹contraction principle and uniqueness of solutions for quasilinear ellipticparabolic equations of the form @ t [b(u)] \Gamma div[a(ru; b(u))] + f(b(u)) = 0 in (0; T )\Theta\Omega ; where b is monotone nondecreasing and continuous. We only assume that u is a weak solution of finite energy (see [1]). In particular, we do not suppose that the distributional derivative @ t [b(u)] is a bounded Borel measure or a locally integrable function.
L¹Contraction And Uniqueness For Unstationary SaturatedUnsaturated Porous Media Flow
"... We prove L¹contraction and thus uniqueness for solutions of the equation modelling unstationary saturatedunsaturated water flow in porous media. Due to three different types of boundary conditions, describing contact with an impervious layer, with water reservoirs of variable level and with the ..."
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We prove L¹contraction and thus uniqueness for solutions of the equation modelling unstationary saturatedunsaturated water flow in porous media. Due to three different types of boundary conditions, describing contact with an impervious layer, with water reservoirs of variable level and with the atmosphere, this ellipticparabolic equation comes as a variational inequality with timedependent constraints at the boundary. The notion of solution considered is the weakest possible: solutions of finite energy. In particular, we do not assume any timeregularity of the saturation s like @ t s 2 L¹(Q) : The proof applies a technique developed by the author for general ellipticparabolic equations.
EXISTENCE AND UNIQUENESS OF SOLUTIONS OF THE BOUSSINESQ SYSTEM WITH NONLINEAR THERMAL DIFFUSION
"... The Boussinesq system of hydrodynamics equations [3], [26] arises from a zero order approximation to the coupling between the Navier–Stokes equations and the thermodynamic equation [25]. Presence of density gradients in a fluid allows the conversion of gravitational potential energy into motion thro ..."
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The Boussinesq system of hydrodynamics equations [3], [26] arises from a zero order approximation to the coupling between the Navier–Stokes equations and the thermodynamic equation [25]. Presence of density gradients in a fluid allows the conversion of gravitational potential energy into motion through the action of buoyant forces. Density gradients are induced, for instance, by temperature differences arising from nonuniform heating of the fluid. In the Boussinesq approximation of a large class of flow problems, thermodynamical coefficients such as viscosity, specific heat and thermal conductivity may be assumed to be constants, leading to a coupled system of parabolic equations with linear second order operators, see, e.g. [11], [12], [17], [31]. However, there are some fluids such as lubrificants or some plasma flow for which this is not an accurate assumption [16], [29] and a quasilinear parabolic system has to be considered. In this paper we present some results on existence and uniqueness of weak solutions for this kind of models. Results on some qualitative properties related with spatial and time localization of the support of solutions will be published elsewhere, see