Related papers: Almost sure local well-posedness for a derivative …

In this paper we continue our study [DSS20] of the nonlinear Schr\"odinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove…

In [12], we proved that $1$-d periodic fractional Schr\"odinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s>\frac{1-\alpha}{2}$ and globally well-posed for $s>\frac{5\alpha-1}{6}$. In this paper we define an…

In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$…

In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity…

This work investigates radial solutions for nonlinear fractional Schr\"odinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the…

We consider the Keller-Segel model for chemotaxis with a nonlinear diffusion coefficent and a singular sensitivity function. We show the existence of travelling waves for wave speeds above a critical value, and establish local…

We study the local well-posedness of the nonlinear Schr\"odinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in the Sobolev spaces $H^k$ when $k \leq…

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz type…

We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schr\"odinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case…

We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we…

We consider the stochastic partial differential equation, $\partial_t u = \tfrac12 \partial^2_x u + b(u) + \sigma(u) \dot{W},$ where $u=u(t\,,x)$ is defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time…

We prove the global well-posedness for a $L^2$-critical defocusing cubic higher-order Schr\"odinger equation, namely \[ i\partial_t u + \Lambda^k u = -|u|^2 u, \] where $\Lambda=\sqrt{-\Delta}$ and $k\geq 3, k \in \mathbb{Z}$ in…

The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane,…

We consider the energy-critical defocusing nonlinear wave equation (NLW) on $\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence and uniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\times…

The deterministic inviscid primitive equations (also called the hydrostatic Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey classes of order strictly greater than 1, and some of their analytic solutions exist only…

We consider the defocusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation (INLS) $iu_t + \Delta u = |x|^{-b}|u|^{k}u$ in $\mathbb{R} \times \mathbb{R}^{n}$ where $n \geq 3$, $0<b<\min(2, n/2)$, and $k=(4-2b)/(n-2)$. We…

We consider the defocusing inhomogeneous nonlinear Schr\"{o}dinger equation $i\partial_tu+\Delta u= |x|^{-b}|u|^{\alpha}u,$ where $0<b<1$ and $0<\alpha<\infty$. This problem has been extensively studied for initial data in $H^1(\R^N)$ with…

The discrete nonlinear Schr\"odinger equation on \(\Z^d\), \(d \geq 1\) is an example of a dispersive nonlinear wave system. Being a Hamiltonian system that conserves also the \(\ell^2(\Z^d)\)-norm, the well-posedness of the corresponding…

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local well-posedness is proved for any $C^2$ power-nonlinearity, while global…

In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space…