Related papers: Decay of weak solutions to the 2D dissipative quas…

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation's wave number dependence is super-linear.…

We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in $L^{2/(\alpha-1)}(\mathbb{R}^2)$. Furthermore, we give upper bound of similar type for any…

We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as…

Califano-Chiuderi \cite{CC} gave the numerical observation that the energy of the MHD equations is dissipated at a rate independent of the ohmic resistivity, which was first proved by \cite{RWXZ}[Ren et al., J. Funct. Anal., 2014] (the…

In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in…

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha< 1/2$) dissipation $(-\Delta)^\alpha$ : If a Leray-Hopf weak solution is H\"{o}lder continuous $\theta\in C^\delta({\mathbb…

We consider the surface quasi-geostrophic equation in two spatial dimensions, with subcritical diffusion (i.e. with fractional diffusion of order $2\alpha$ for $\alpha>\frac{1}{2}$.) We establish existence of solutions without assuming…

We consider the temporal decay estimates for weak solutions to the two-dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non-uniform decay \begin{align*} \|u(t)\|_{L^{2}}+\|\nabla…

In this paper we study the behavior of the energy of solutions of the wave equation with localized damping in exterior domain. We assume that the damper is positive at infinity. Under the Geometric Control Condition of Bardos et al (1992),…

It has been known that if the initial data decay sufficiently fast at space infinity, then 1D Klein-Gordon equations with quadratic nonlinearity admit classical solutions up to time $e^{C/\epsilon^2}$ while $e^{C/\epsilon^2}$ is also the…

We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in $\mathbb{R}^3$ when $H^3$ norm of the initial data is small. If further, the $\dot{H}^{-s}$ norm ($0\leq s<3/2)$ or…

We prove global existence, uniqueness and stability of entropy solutions with $L^2\cap L^\infty$ initial data for a general family of negative order dispersive equations. It is further demonstrated that this solution concept extends in a…

We first construct the global unique solution by assuming that the initial data is small in the H^3 norm but its higher order derivatives could be large. If further the initial data belongs to \Dot{H}^{-s} (0\le s<3/2) or…

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\bf S}_\alpha)$. We prove that there exists a unique local-in-time solution for any large initial data $\theta_0$ in the space ${\bf{\mathcal…

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in $ \mathbb{R}^2$. Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data…

We prove a large-data $L^2$-decay estimate for nonlinear dissipative Schr\"odinger equations with attractive-dissipative power nonlinearity. The main difficulty is the lack of sign definiteness of the standard energy when $\Re\lambda<0$,…

We study the initial-boundary value problem for 1D compressible MHD equations of viscous non-resistive fluids in the Lagrangian mass coordinates. Based on the estimates of upper and lower bounds of the density, weak solutions are…

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove global well-posedness result for small initial data lying in critical Besov spaces constructed…

In this paper, we are concerned with possibly degenerate stochastic partial differential equations (SPDEs). An $L^2$-theory is introduced, from which we derive the H\"ormander theorem with an analytical approach. With the method of De…

We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $\kappa(-\Delta)^{\alpha/2}\theta\ (0<\alpha\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a…