DCDS-S
The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity
Madalina Petcu Roger Temam
Discrete & Continuous Dynamical Systems - S 2011, 4(1): 209-222 doi: 10.3934/dcdss.2011.4.209
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.
keywords: well posedness. a priori estimates shallow water
DCDS
Renormalization group method: Application to Navier-Stokes equation
I. Moise Roger Temam
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 191-210 doi: 10.3934/dcds.2000.6.191
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.
keywords: Renormalization group method Galerkin method. Navier­Stokes equations
DCDS
An interface problem: The two-layer shallow water equations
Madalina Petcu Roger Temam
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5327-5345 doi: 10.3934/dcds.2013.33.5327
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.
keywords: Shallow water equations transparent boundary conditions two layers fluid strictly dissipative boundary conditions.
DCDS-B
The linear hyperbolic initial and boundary value problems in a domain with corners
Aimin Huang Roger Temam
Discrete & Continuous Dynamical Systems - B 2014, 19(6): 1627-1665 doi: 10.3934/dcdsb.2014.19.1627
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.
keywords: Hyperbolic partial differential equations simultaneous congruence diagonalization initial and boundary value problem semigroup.
DCDS-B
Exponential approximations for the primitive equations of the ocean
Roger Temam D. Wirosoetisno
Discrete & Continuous Dynamical Systems - B 2007, 7(2): 425-440 doi: 10.3934/dcdsb.2007.7.425
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.
keywords: exponential asymptotics Gevrey regularity primitive equations Singular perturbation
DCDS-B
Existence theorem for a model of dryland vegetation
Yukie Goto Danielle Hilhorst Ehud Meron Roger Temam
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 197-224 doi: 10.3934/dcdsb.2011.16.197
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.
keywords: porous media equations Desertification Galerkin approximation parabolic equations degenerate parabolic equations Dryland vegetation compactness theorems.
CPAA
The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction
Aimin Huang Roger Temam
Communications on Pure & Applied Analysis 2014, 13(5): 2005-2038 doi: 10.3934/cpaa.2014.13.2005
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.
keywords: existence and uniqueness. hyperbolic equations Shallow water equations
DCDS
Interaction of boundary layers and corner singularities
Chang-Yeol Jung Roger Temam
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 315-339 doi: 10.3934/dcds.2009.23.315
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}.
keywords: boundary layers; boundary layer element; finite elements; singularly perturbed problem; convection-diffusion
DCDS
Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach
Jesus Ildefonso Díaz David Gómez-Castro Jean Michel Rakotoson Roger Temam
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 509-546 doi: 10.3934/dcds.2018023

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.

keywords: Linear diffusion equations singular absorption potential unbounded convective flow no boundary conditions dual problem local Kato inequality distance to the boundary weighted spaces
DCDS
Mark Vishik and his work
Roger Temam
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): i-vi doi: 10.3934/dcds.2004.10.1i
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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