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Finite-dimensional Pullback Attractors for Non-autonomous Newton-Boussinesq Equations in Some Two-dimensional Unbounded Domains

Cung The Anh, Dang Thanh Son (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We study the existence and long-time behavior of weak solutions to Newton-Boussinesq equations in two-dimensional domains satisfying the Poincaré inequality. We prove the existence of a unique minimal finite-dimensional pullback D σ -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms.

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Robert Černý (2014)

Open Mathematics

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem u W 0 1 L Φ Ω a n d - d i v Φ ' u u u + V x Φ ' u u u = f x , u + μ h x i n Ω , where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on N

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove the existence of global attractors for the following semilinear degenerate parabolic equation on N : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method.

Global strong solutions of a 2-D new magnetohydrodynamic system

Ruikuan Liu, Jiayan Yang (2020)

Applications of Mathematics

The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic model in the two dimensional space. Based on Agmon, Douglis, and Nirenberg’s estimates for the stationary Stokes equation and Solonnikov’s theorem on L p - L q -estimates for the evolution Stokes equation, it is shown that this coupled magnetohydrodynamic equations possesses a global strong solution. In addition, the uniqueness of the global strong solution is obtained.

Growing Sobolev norms for the cubic defocusing Schrödinger equation

Zaher Hani, Benoit Pausader, Nikolay Tzvetkov, Nicola Visciglia (2013/2014)

Séminaire Laurent Schwartz — EDP et applications

This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles, Emmanuel Chasseigne, Cyril Imbert (2011)

Journal of the European Mathematical Society

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach

Abdellatif Benchérif-Madani, Étienne Pardoux (2007)

ESAIM: Probability and Statistics

In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.

Integrability for solutions to quasilinear elliptic systems

Francesco Leonetti, Pier Vincenzo Petricca (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we prove an estimate for the measure of superlevel sets of weak solutions to quasilinear elliptic systems in divergence form. In some special cases, such an estimate allows us to improve on the integrability of the solution.

Integrability for very weak solutions to boundary value problems of p -harmonic equation

Hongya Gao, Shuang Liang, Yi Cui (2016)

Czechoslovak Mathematical Journal

The paper deals with very weak solutions u θ + W 0 1 , r ( Ω ) , max { 1 , p - 1 } < r < p < n , to boundary value problems of the p -harmonic equation - div ( | u ( x ) | p - 2 u ( x ) ) = 0 , x Ω , u ( x ) = θ ( x ) , x Ω . ( * ) We show that, under the assumption θ W 1 , q ( Ω ) , q > r , any very weak solution u to the boundary value problem ( * ) is integrable with u θ + L weak q * ( Ω ) for q < n , θ + L weak τ ( Ω ) for q = n and any τ < , θ + L ( Ω ) for q > n , provided that r is sufficiently close to p .

Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case q = 3 d d + 2

Jörg Wolf (2007)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider weak solutions 𝐮 : Ω d to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain Ω d ( d = 2 or d = 3 ). For the critical case q = 3 d d + 2 we prove the higher integrability of 𝐮 which forms the basis for applying the method of differences in order to get fractional differentiability of 𝐮 . From this we show the existence of second order weak derivatives of u .

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