1,078 research outputs found
On uniqueness techniques for degenerate convection-diffusion problems
International audienceWe survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form u_t-\div\, (a_0(\Grad w)+F(w))=f, for a maximal monotone graph , a Leray-Lions type nonlinearity , a continuous convection flux , and an initial condition . The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space. We avoid the degeneracy that could make the problem hyperbolic in some regions; yet our starting point is the notion of entropy solution, notion that underlies the theory of general hyperbolic-parabolic-elliptic problems. Thus, we focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the ''parabolic-elliptic aspects''. We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of ''going to the boundary'' in the Kato inequality for comparison of two solutions; uniqueness for renormalized solutions obtained via reduction to weak solutions. On several occasions, the results are achieved thanks to the notion of integral solution coming from the nonlinear semigroup theory
Degenerate parabolic equation with zero flux boundary condition and its approximations
We study a degenerate parabolic-hyperbolic equation with zero flux boundary
condition. The aim of this paper is to prove convergence of numerical
approximate solutions towards the unique entropy solution. We propose an
implicit finite volume scheme on admissible mesh. We establish fundamental
estimates and prove that the approximate solution converge towards an
entropy-process solution. Contrarily to the case of Dirichlet conditions, in
zero-flux problem unnatural boundary regularity of the flux is required to
establish that entropy-process solution is the unique entropy solution. In the
study of well-posedness of the problem, tools of nonlinear semigroup theory
(stationary, mild and integral solutions) were used in [Andreianov, Gazibo,
ZAMP, 2013] in order to overcome this difficulty. Indeed, in some situations
including the one-dimensional setting, solutions of the stationary problem
enjoy additional boundary regularity. Here, similar arguments are developed
based on the new notion of integral-process solution that we introduce for this
purpose.Comment: 41 page
contraction for bounded (non-integrable) solutions of degenerate parabolic equations
We obtain new contraction results for bounded entropy solutions of
Cauchy problems for degenerate parabolic equations. The equations we consider
have possibly strongly degenerate local or non-local diffusion terms. As
opposed to previous results, our results apply without any integrability
assumption on the %(the positive part of the difference of) solutions. They
take the form of partial Duhamel formulas and can be seen as quantitative
extensions of finite speed of propagation local contraction results for
scalar conservation laws. A key ingredient in the proofs is a new and
non-trivial construction of a subsolution of a fully non-linear (dual)
equation. Consequences of our results are maximum and comparison principles,
new a priori estimates, and in the non-local case, new existence and uniqueness
results
Well-posedness results for triply nonlinear degenerate parabolic equations
We study the well-posedness of triply nonlinear degenerate
elliptic-parabolic-hyperbolic problem in a bounded domain with
homogeneous Dirichlet boundary conditions. The nonlinearities and
are supposed to be continuous non-decreasing, and the nonlinearity
falls within the Leray-Lions framework. Some restrictions
are imposed on the dependence of on
and also on the set where degenerates. A model case is
with which is strictly increasing except on a locally finite number of
segments, and which is of the Leray-Lions kind. We are
interested in existence, uniqueness and stability of entropy solutions. If
, we obtain a general continuous dependence result on data
and nonlinearities . Similar result
is shown for the degenerate elliptic problem which corresponds to the case of
and general non-decreasing surjective . Existence, uniqueness
and continuous dependence on data are shown when and
is continuous
Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise
In this article, we are concerned with a multidimensional degenerate
parabolic-hyperbolic equation driven by Levy processes. Using bounded variation
(BV) estimates for vanishing viscosity approximations, we derive an explicit
continuous dependence estimate on the nonlinearities of the entropy solutions
under the assumption that Levy noise depends only on the solution. This result
is used to show the error estimate for the stochastic vanishing viscosity
method. In addition, we establish fractional BV estimate for vanishing
viscosity approximations in case the noise coefficients depend on both the
solution and spatial variable.Comment: 31 Pages. arXiv admin note: text overlap with arXiv:1502.0249
Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
We study a class of degenerate convection diffusion equations with a
fractional nonlinear diffusion term. These equations are natural
generalizations of anomalous diffusion equations, fractional conservations
laws, local convection diffusion equations, and some fractional Porous medium
equations. In this paper we define weak entropy solutions for this class of
equations and prove well-posedness under weak regularity assumptions on the
solutions, e.g. uniqueness is obtained in the class of bounded integrable
functions. Then we introduce a monotone conservative numerical scheme and prove
convergence toward an Entropy solution in the class of bounded integrable
functions of bounded variation. We then extend the well-posedness results to
non-local terms based on general L\'evy type operators, and establish some
connections to fully non-linear HJB equations. Finally, we present some
numerical experiments to give the reader an idea about the qualitative behavior
of solutions of these equations
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws
We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes
- …