Related papers: Ill-Posedness of the Third Order NLS Equation with…

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…

We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical…

This work is concerned with the Cauchy problem for a coupled Schr\"odinger-Benjamin-Ono system $$\left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv,\qquad t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+\nu\mathcal…

This paper investigates the Cauchy problem for the nonlinear Schr\"odinger equation (NLS) in the mass-supercritical and energy-subcritical regime within three spatial dimensions. For initial data in the critical homogeneous Sobolev space…

We consider the Cauchy problem for the 2D and 3D Klein-Gordon-Schr\"odinger system. In 2D we show local well-posedness for Schr\"odinger data in H^s and wave data in H^{\sigma} x H^{\sigma -1} for s=-1/4 + and \sigma = -1/2, whereas…

We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u=0 \end{align*} in the…

Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for…

We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for 3D compressible Navier-Stokes-Allen-Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the…

This paper is mainly concerned with the well-posedness and exponential decay of solution for a integrable three-component Novikov system, which admits bi-Hamiltonian structure and infinitely many conserved quantities. The local…

In this paper, we study the Cauchy problem of a higher-order $\mu$-Camassa-Holm equation. By employing the Green's function of $(\mu-\partial_{x}^{2})^{-2}$, we obtain the explicit formula of the inverse function…

We consider the Cauchy problem for a quadratic derivative nonlinear Schr\"odinger equation whose nonlinearity is a linear combination of $\partial_x (u^2)$ and $\partial_x (|u|^2)$. We prove the local well-posedness in the $L^2$-based…

In this paper we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a…

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n+2)-dimensional static and spherically symmetric spacetimes. They are related to properties of the…

We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in…

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the…

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $\hat{H}^r_s(\R)$ defined by the norm $$\n{v_0}{\hat{H}^r_s(\R)} := \n{< \xi > ^s\hat{v_0}}{L^{r'}_{\xi}},\quad < \xi…

We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t \omega + u\cdot\nabla\omega = R(\omega),\] where $R$ is any non-trivial second order Riesz transform. Namely, we…

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the…

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…