DCDS-S

In the present article we consider the nonviscous Shallow Water Equations in space dimension one with
Dirichlet boundary conditions for the velocity and we
show the locally in time well-posedness of the model.

DCDS

The aim of this article is to present a rather unusual and partly heuristic
application of the renormalization group (RG) theory to the Navier-Stokes equations
with space periodic boundary conditions.
We obtain in this way a new nonlinear
renormalized equation with a nonlinear term which is invariant under the Stokes
operator.
Its relation to the Navier-Stokes equations is investigated for non-resonant
domains.

DCDS

The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.

DCDS-B

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.

DCDS-B

We show that in the limit of small Rossby number $\varepsilon$, the primitive
equations of the ocean (OPEs) can be approximated by "higher-order
quasi-geostrophic equations'' up to an exponential accuracy in $\varepsilon$.
This approximation assumes well-prepared initial data and is valid for
a timescale of order one (independent of $\varepsilon$).
Our construction uses Gevrey regularity of the OPEs and a classical
method to bound errors in higher-order perturbation theory.

DCDS-B

In this article, we consider the dryland vegetation model proposed
by Gilad et al [6, 7]. This model
consists of three nonlinear parabolic partial differential
equations, one of which is degenerate parabolic of the family
of the porous media equation [3, 7],
and we prove the existence of its weak solutions. Our approach based
on the classical Galerkin methods combines and makes use of techniques,
parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of
the solutions.

CPAA

In continuation with earlier works on the shallow water equations in a rectangle [10, 11], we investigate in this article the fully inviscid nonlinear shallow water equations in space dimension two in a rectangle $(0,1)_x \times (0,1)_y$. We address in this article the subcritical case, corresponding to the condition (3) below. Assuming space periodicity in the $y$-direction, we propose the boundary conditions for the $x$-direction which are suited for the subcritical case and develop, for this problem, results of existence, uniqueness and regularity of solutions locally in time for the corresponding initial and boundary value problem.

DCDS

Our aim in this article is to study the interaction of
\textit{boundary layers} and \textit{corner singularities} in the
context of singularly perturbed convection-diffusion equations. For
the problems under consideration, we determine a simplified form of
the corner singularities and show how to use it for the numerical
approximation of such problems in the context of variational
approximations using the concept of \textit{enriched spaces}.

DCDS

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\text{IR}^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

DCDS

The name of Mark Vishik is one of the very first names that I heard
when I started research in mathematics in 1964. One year before in
1963, Mark published an article that had a very deep influence on the
theory of nonlinear partial differential equations all along the
1960s, although this paper is nearly forgotten by now, and probably
very few know about it. Later on I will describe in detail this part
of his career that I witnessed during the preparation of my
thesis.

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