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We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…
We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\pm |x|^{-b}|u|^{\frac{4-2b}{d}}u$ for large $L^2(\mathbb{R} ^d)$ initial data with…
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.…
We present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation.…
We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive…
In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in $H^{s}$ with spatial dimension $n \leq 5$. We show this equation, with power $2\le p\le 1+4/(n-1)$, is…
In this paper, we study the long time well-posedness for the nonlinear Prandtl boundary layer equation on the half plane. While the initial data are small perturbations of some monotonic shear profile, we prove the existence, uniqueness and…
In this paper we study the inhomogeneous incompressible Euler equations in the whole space $\mathbb{R}^n$ with $n\geq3$. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov-Herz…
In this paper, the well-posedness is studied for the initial boundary value problem of the two-dimensional compressible ideal magnetohydrodynamic (MHD) equations in bounded perfectly conducting domains with corners. The presence of corners…
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right.…
In the literature on singular perturbation (Lavrentiev regularization) for the stable approximate solution of operator equations with monotone operators in the Hilbert space the phenomena of conditional stability and local well-posedness…
In this paper, we investigate the well-posedness of the martingale problem associated to non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov under mild assumptions on the coefficients as well as classical…
We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schr\"odinger equations with L\'{e}vy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems…
We consider a class of elliptic variational-hemivaria\-tional inequalities in a abstract Banach space for which we introduce the concept of well-posedness in the sense of Tykhonov. We characterize the well-posedness in terms of metric…
We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s > -1/2$, for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type spaces. This result is almost…
We investigate the initial value problem for some energy supercritical semilinear wave equations. We establish local existence in suitable spaces with continuous flow. We also obtain some ill-posedness/weak ill-posedness results. The proof…
We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$\alpha$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $\alpha$ $\le$ 1,…
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…
In this paper we study the well-posedness for the inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in Sobolev spaces $H^s$, $s\geq0$. The well-posedness theory for this model has been…
We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $…