Related papers: Probabilistic well-posedness for supercritical wav…

We show new global well-posedness results for mass-critical nonlinear Schr\"odinger equations on tori in one and two dimensions. For the quintic nonlinear Schr\"odinger equation on the circle we show global well-posedness for initial data…

In this paper, we show the global well-posedness for periodic gKdV equations in the space $H^s(\mathbb{T})$, $s\ge \frac12$ for quartic case, and $s> \frac59$ for quintic case. These improve the previous results of I-team in 2004. In…

We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial…

The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in $H^s({\mathbb{R}^2})$ for all $\frac{5}{7}<s<1$ via using $I$-method in the context of atomic spaces. By means of the increment…

This paper deals with a class of semilinear wave equation with nonlinear damping term $|u_{t}|^{m-2}u_t $ and nonlinear source term $g(x)|u|^{p-2}u$ on the manifolds with conical singularities. Firstly, we prove the local existence and…

This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing…

We consider the wave and Klein-Gordon equations on the real hyperbolic space $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of…

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…

For the radial energy-supercritical nonlinear wave equation $$\Box u = -u_{tt} + \triangle u = \pm u^7$$ on $\R^{3+1}$, we prove the existence of a class of global in forward time $C^\infty$-smooth solutions with infinite critical Sobolev…

In this paper we prove global well-posedness and scattering for the defocusing, intercritical nonlinear wave equation in dimensions $d \geq 4$ with radial initial data. We prove this for sharp initial data.

We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on $\mathbb{T}^2$. We prove the local well-posedness for given data in $H^s(\mathbb{T}^2)$ whenever $s>5/3$. More importantly, we prove that this equation…

We prove local well-posedness of partially periodic and periodic modified KP-I equations, namely for $\partial_t u+(-1)^{\frac{l+1}{2}}\partial^l_x u-\partial_x^{-1}\partial_y^2 u+u^2\partial_x u=0$ in the anisotropic Sobolev space…

The Cauchy Problem for the modified Zakharov-Kuznetsov equation in three space dimensions is shown to be locally well-posed in $H^s(\R^3)$ for $s > \frac12$. Combined with the conservation of mass and energy this result implies global…

In this manuscript, we focus on the more delicate nonlinearity of the semilinear wave equation $$\partial_{t}^2 u-\Delta_{\mathbb{R}^3}u=|u|^{p_S}\mu(|u|)\ ,u(0,x)=\varepsilon u_0,\ u_t(0,x)=\varepsilon u_1\ ,$$ where $p_S=1+\sqrt{2}$ is…

This paper introduces a novel class of initial data for which the three-dimensional incompressible Navier--Stokes equations yield unique global-in-time solutions. Building on a logarithmically improved regularity criterion, we impose a…

In this article, we consider the global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity. More precisely, assuming $a_0 \in \dot{B}_{q,1}^{\frac{3}{q}}(\mathbb{R}^3)$ and…

We discuss how the higher-order term $|u|^q$ $(q>1+2/(n-1))$ has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations $$ \partial_t^2 u-\Delta u=|v|^p, \qquad \partial_t^2…

In this paper, we prove the existence of a countable family of regular spherically symmetric self-similar solutions to focusing energy super-critical semi-linear wave equations \begin{equation*} \partial_{tt}u-\Delta u=|u|^{p-1}u \qquad…

In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay…

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation $\partial_tu+\Delta\partial_xu+ u\partial_xu=0$ in the Sobolev spaces $H^s(\R^3)$, $s>1$, as well as in the Besov space $B^{1,1}_2(\R^3)$. The proof is…