Related papers: On Gibbs measure and weak flow for the cubic NLS w…

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first…

We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $\gamma\ge1$. Initial data and solutions are small in $L^2$ around a…

We study the mixed local and nonlocal double phase parabolic equation \begin{align*} \partial_t u(x,t)-\mathrm{div}(a(x,t)|\nabla u|^{q-2}\nabla u) +\mathcal{L}u(x,t)=0 \end{align*} in $Q_T=\Omega\times(0,T)$, where $\mathcal{L}$ is the…

We study a complex non-newtonian fluid that models the flow of nematic liquid crystals. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. We prove the existence of global weak…

This paper presents a mathematical analysis of a one-dimensional model of turbulence based on a stochastic generalized Constantin-Lax-Majda-DeGregorio (gCLMG) equation. We focus on the specific case where the nonlinearity in the equation…

We establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak Gibbs measures. Under some mild assumptions, we obtain upper and lower bounds for the measure of…

We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all}…

We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's…

For a nonlinear operator $T$ satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) $T$ is surjective, ii) $T$ is open at zero, and iii) $T$ has a bounded right inverse. The theorem…

We utilize Gaussian measure preserving systems to prove the existence and genericity of Lebesgue measure preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically…

In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{T}$ from many-body quantum Gibbs states. On the quantum many-body level, the quintic…

We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…

We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a…

In this paper, we study the nonlinear modulational instability of two-dimensional hydroelastic Stokes waves in infinite depth. We first justify a focusing cubic nonlinear Schr\"odinger (NLS) approximation result for 2D deep hydroelastic…

Given any $\mu_1, \mu_2\in {\mathbb C}$ and $\alpha >0$, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $\partial_{ tt } u - \Delta u + \mu_1 u = \mu_2 |u|^\alpha u$ on ${\mathbb R}^N$,…

We consider the log-perturbed Br\'ezis-Nirenberg problem on the hyperbolic space \begin{align*} \Delta_{\mathbb{B}^N}u+\lambda u +|u|^{p-1}u+\theta u \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N,…

In this paper we consider the It\^o SDE $$d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in {\mathbb R}^d,$$ where $W_t$ is a $d$-dimensional standard Wiener process and the drift coefficient $b:[0,T]\times{\mathbb R}^d\to{\mathbb R}^d$ belongs to…

In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schr\"{o}dinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear…

We investigate the incompressible Navier-Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the initial…