Related papers: Polynomial Bound and Nonlinear Smoothing for the B…
We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $\mathbb{T}$ with initial data below $L^{2}(\mathbb{T})$. With respect to random initial data of strictly negative Sobolev…
In this paper we deal with the initial value problem related to a family of dispersive inhomogeneous evolution equations Pu=f with variable coefficients belonging to the class of p-evolution equations, $p\geq 2$. We study the smoothing…
In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the Cauchy problem in local Sobolev…
We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order…
The Cauchy problem of the modified nonlinear Schr\"{o}dinger (mNLS) equation with the finite density type initial data is investigated via $\overline{\partial}$ steepest descent method. In the soliton region of space-time $x/t\in(5,7)$, the…
In this paper, we consider the 3D Jordan--Moore--Gibson--Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are…
The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({\mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of…
In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for…
We examine the solution of the Benjamin-Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one…
In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary…
In this manuscript, we would established in low regularity spaces $H^\ell, \ell\in [0,1)$, the existence and stability results of time-periodic solution of 1D Cauchy problem of forced damped Benjamin-Bona-Mahony equation (BBM). We use…
We consider the cubic nonlinear Schr\"odinger equation on product manifolds $\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of the solutions. The main ingredient of the…
We prove that the smooth solutions to the Cauchy problem for the three-dimensional compressible barotropic magnetohydrodynamic equations with conserved total mass and finite total energy lose the initial smoothness within a finite time.…
Having the ill-posedness in the range $s<-3/4$ of the Cauchy problem for the Benjamin equation with an initial $H^{s}({\mathbb R})$ data, we prove that the already-established local well-posedness in the range $s>-3/4$ of this initial value…
We consider semilinear equations of the form p(D)u=F(u), with a locally bounded nonlinearity F(u), and a linear part p(D) given by a Fourier multiplier. The multiplier p(\xi) is the sum of positively homogeneous terms, with at least one of…
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…
In this paper we study the local and global regularity properties of the cubic nonlinear Schr\"odinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and…
We consider the nonlinear Schrodinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse: there is a loss of regularity,…
In this paper, the finite time blow-up of smooth solutions to the Cauchy problem for full Euler-Poisson equations and isentropic Euler-Poisson equations with repulsive forces or attractive forces in high dimensions $(n\geq3)$ is proved for…
We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove…