S. N. Antontsev

The Navier–Stokes problem modified by an absorption term

It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|σ−2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|σ−2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 < σ < 2 and exponentially decay in time if σ = 2. In the special case of a suitable force field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided 1 < σ < 2. We also prove that for non-zero body forces decaying at a power-time rate, the solutions decay at analogous power-time rates if σ > 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.

The Euler-Galerkin finite element method for a nonlocal coupled system of reaction-diffusion type

In this work, we study a system of parabolic equations with nonlocal nonlinearity of the following type { u t - a 1 ( l 1 ( u ) , l 2 ( v ) ) Δ u + λ 1 | u | p - 2 u = f 1 ( x , t ) in? ? × 0 , T v t - a 2 ( l 1 ( u ) , l 2 ( v ) ) Δ v + λ 2 | v | p - 2 v = f 2 ( x , t ) in? ? × 0 , T u ( x , t ) = v ( x , t ) = 0 on? ? ? × 0 , T u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) in? ? , where a 1 and a 2 are Lipschitz-continuous positive functions, l 1 and l 2 are continuous linear forms, λ 1 , λ 2 ? 0 and p ? 2 .We prove the convergence of a linearized Euler-Galerkin finite element method and obtain the order of convergence in the L 2 norm. Finally we implement and simulate the presented method in Matlabs environment.

Higher regularity of solutions of singular parabolic equations with variable nonlinearity

ABSTRACT We study the global regularity of solutions of the homogeneous Dirichlet problem for the parabolic equation with variable nonlinearity where p(x, t), are given functions of their arguments, and . Conditions on the data are found that guarantee the existence of a unique strong solution such that and . It is shown that if with , p and are Hölder-continuous in , and , then for every strong solution with any .

Finite Speed of Propagation and Waiting Time for Local Solutions of Degenerate Equations in Viscoelastic Media or Heat Flows with Memory

The finite speed of propagation (FSP) was established for certain materials in the 70’s by the American school (Gurtin, Dafermos, Nohel, etc.) for the special case of the presence of memory effects. A different approach can be applied by the construction of suitable super and sub-solutions (Crandall, Nohel, Díaz and Gomez, etc.). In this paper we present an alternative method to prove (FSP) which only uses some energy estimates and without any information coming from the characteristics analysis. The waiting time property is proved for the first time in the literature for this class of nonlocal equations.

Singular Perturbations of Forward-Backward p-Parabolic Equations

In this paper we have proved the existence of entropy measure-valued solutions to forward-backward p-parabolic equations. We have obtained these solutions as singular limits of weak solutions to (p,q)-elliptic regularized boundary-value problems as ε → 0+. When q > 1 and q = 2 we have not defined yet admissible initial and final conditions even in the form of integral inequalities.

On the localization of solutions of doubly nonlinear parabolic equations with nonstandard growth in filtration theory

We study the properties of space localization of weak solutions of the equation which appears in the mathematical description of filtration of an ideal barotropic gas in a porous medium. The functions and are assumed to satisfy the nonstandard growth conditions: , , , , with some positive constants and measurable bounded functions , , . It is shown that if , , and , meet certain regularity requirements, then every weak solution possesses the property of finite speed of propagation of disturbances from the initial data. In the case that in a ball and in , the solutions display the waiting time property: if with a positive exponent , depending on and , and a sufficiently small , then there exists such that in .