Related papers: Smoothing effect for higher order dispersive equat…
We analyze the effect of pressure on the evolution of perturbations of an Einstein-de Sitter Universe in the matter dominated epoch assuming an ideal gas equation of state. For the sake of simplicity the temperature is considered uniform.…
We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This…
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
We study the question of existence of time-periodic, spatially periodic solutions for dispersive evolution equations, and in particular, we introduce a framework for demonstrating the nonexistence of such solutions. We formulate the problem…
We study an initial value problem for two-dimensional needle crystal growth with anisotropic surface tension. The initial value problem is derived from the so called one-sided model based on complex variables method. We then obtain the…
We prove some local smoothing estimates for the Schr\"{o}dinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay…
We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $|x|$…
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$…
We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schr\"odinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both…
This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper, where dispersive equations were treated.…
Ill posed linear and nonlinear initial value problems may be stabilized, that it converted to to well posed initial value problems, by the addition of purely nonscalar linear dispersive terms. This is a stability analog of the Turing…
We investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general…
We investigate the evolution of localized initial value profiles when propagated in integrable versions of higher time-derivative theories. In contrast to the standard cases in nonlinear integrable systems, where these profiles evolve into…
The article first studies the propagation of well prepared high frequency waves with small amplitude $\eps$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating…
In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same…
We study regularity and decay properties for the solutions of the Cauchy problem for time-fractional partial differential equations, with tempered initial data, belonging to suitable (weighted) Sobolev spaces, associated with a differential…
We study the Cauchy problem of the Kolmogorov-Fokker-Planck equations and show that the solution enjoys an analytic smoothing effect with L2 initial datum for positive time.
For generalized KdV models with polynomial nonlinearity, we establish nonlinear smoothing property in $H^s$ for $s>\frac{1}{2}$. Such smoothing effect persists globally, provided that the $H^1$ norm does not blow up in finite time. More…
Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning the linearized problem, the decay estimate with diffusion wave property for an initial data is derived. As an application, the time…