Related papers: Local wellposedness for the 2+1 dimensional monopo…

We consider the Cauchy problem for an equation of the form \partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms and no quadratic uu_{xx} term. For a polynomial nonlinearity with no quadratic…

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schr\"odinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of…

In this paper, we study local well-posedness theory of the Cauchy problem for Schr\"{o}dinger-KdV system in Sobolev spaces $H^{s_1}\times H^{s_2}$. We obtain the local well-posedness when $s_1\geq 0$, $\max\{-3/4,s_1-3\}\leq s_2\leq…

The Zakharov system in dimension $d\leqslant 3$ is shown to be locally well-posed in Sobolev spaces $H^s \times H^l$, extending the previously known result. We construct new solution spaces by modifying the $X^{s,b}$ spaces, specifically by…

In this article, we prove the global well-posedness in the critical Sobolev space $H_{rad}^2\left(\mathbb{R}^2\right) \times H_{rad}^1 \left(\mathbb{R}^2\right)$ for the radial time-like extremal hypersurface equation in $\left(1+3\right)$-…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit…

We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique…

We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence…

We prove the local well-posedness for a two phase problem of magnetohydrodynamics with a sharp interface. The solution is obtained in the maximal regularity space: $H^1_p((0, T), L_q) \cap L_p((0, T), H^2_q)$ with $1 < p, q < \infty$ and…

We study the local well-posedness theory for the Schr\"odinger Maps equation. We work in $n+1$ dimensions, for $n \geq 2$, and prove a local well-posedness for small initial data in $H^{\frac{n}{2}+\e}$.

By using a bilinear smoothing estimate recently developed in [12], together with several linear Strichartz-type estimates established therein, we improve the threshold for local well-posedness of the quartic Zakharov-Kuznetsov equation and…

We establish local well-posedness of the Hall-magneto-hydrodynamics (Hall-MHD) system in the Sobolev space $\left(H^s(\mathbb{R}^n)\right)^2$ with $s>\frac n2$. The previously known local well-posedness space was…

We study the Yang-Mills-Higgs system within the framework of general relativity. In the static situation, using Bogomol'nyi type analysis, we derive a positive-definite energy functional which has a lower bound. Specializing to the gauge…

The Cauchy problem for the Chern-Simons-Higgs system in the (2+1)-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in…

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in $H^s $ with $s>\frac14$. The equation is $L^2$-critical, and hence a $\frac14$ derivative gap is left between…

We prove local well-posedness results for the semi-linear wave equation for data in $H^\gamma$, $0 < \gamma < \frac{n-3}{2(n-1)}$, extending the previously known results for this problem. The improvement comes from an introduction of a…

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…

We show that the viscous resistive magneto-hydrodynamics system with Hall effect is locally well-posed in $H^s(\mathbb R^n)\times H^{s+1-\varepsilon}(\mathbb R^n)$ with $s>\frac{n}2-1$ and any small enough $\varepsilon>0$ such that…

We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which…

For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem.…