Nonlinear Analysis-Theory Methods & Applications最新文献

In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovich noise, as well as nonlinear colored Wiener noise. By means of Trotter–Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.

In this paper, we establish a necessary and sufficient condition for the solvability of the real $(n-1)$ Monge–Ampère equation $\stackrel{1/n}{det}(\Delta uI-D^{2}u)=g(x,u)$ in bounded domains with infinite Dirichlet boundary condition. The $(n-1)$ Monge–Ampère operator is derived from geometry and has recently received much attention. Our result embraces the case $g(x,u)=h\left(x\right)f\left(u\right)$ where $h\in C^{\infty }\left(\bar{\Omega }\right)$ is positive and $f$ satisfies the Keller–Osserman type condition. We describe the asymptotic behavior of the solution by constructing suitable sub-solutions and super-solutions, and obtain a uniqueness result in star-shaped domains by using a scaling technique.

We study the minimizers of a degenerate case of the Ohta–Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.

The purpose of this paper is to provide a class of large initial data which generates global solutions of the compressible flow of liquid crystals in $R^3$. This class of data relax the smallness restriction imposed on the initial incompressible velocity. Moreover, the result improve considerably the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), 2678-2699].

In this paper, we consider the 1D Euler equation with time and space dependent damping term $-a(t,x)v$. It has long been known that when $a(t,x)$ is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient $a$ satisfies the following condition $|a(t,x)-{\mu }_0|\le a_1\left(t\right)+a_2\left(x\right),$where ${\mu }_0\ge 0$ and $a_1$ and $a_2$ are integrable functions with $t$ and $x$. Under this condition, we show the global existence and the blow-up with small initial data, when ${\mu }_0>0$ and ${\mu }_0=0$ respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.

We consider the initial value problem associated to the low dispersion fractionary Benjamin–Bona–Mahony equation, fBBM. Our aim is to establish local persistence results in weighted Sobolev spaces and to obtain unique continuation results that imply that those results above are sharp. Hence, arbitrary polynomial type decay is not preserved by the fBBM flow.

We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.