Attractors for partly dissipative lattice dynamic systems in l2×l2

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in $l^2\times l^2$. We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in $l^2\times l^2$. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.     

Decay of solutions to a new Hall–MHD system in R3

This paper discusses the large-time behavior of solutions for a new Hall–MHD system in ${\mathbb{R}}^3$. Using the Fourier splitting method, we establish the upper bound of the time-decay rate in $L^2({\mathbb{R}}^3)$ for weak solutions.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

Complicated asymptotic behavior of solutions for porous medium equation in unbounded space

In this paper, we find that the unbounded spaces $Y_{\sigma }({\mathbb{R}}^N)$$(0<\sigma <\frac{2}{m?1})$ can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted $L^1$–$L^{\infty }$ estimates for the solutions.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

The Camassa–Holm equation as an incompressible Euler equation: A geometric point of view

The group of diffeomorphisms of a compact manifold endowed with the $L^2$ metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein–Fisher–Rao distance is a natural extension of the classical $L^2$-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the $H^{\mathrm{div}}$ right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa–Holm (CH) equation in one dimension. Geometrically, we present an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an $L^2$ type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On $S_1$, solutions to the standard CH thus give radially 1-homogeneous solutions of the incompressible Euler equation on ${\mathbb{R}}^2$ which preserves a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler–Arnold equation for the $H^{\mathrm{div}}$ right-invariant metric are length minimizing geodesics for sufficiently short times.     

Convergence rates of Neumann problems for Stokes systems

In quantitative homogenization of the Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, this paper studies the convergence rates of the velocity in $L^2$ and $H^1$ as well as those of the pressure term in $L^2$, without any smoothness assumptions on the coefficients.     

Decay rates of solutions to a P1-approximation model arising from radiation hydrodynamics

This paper is concerned with the global existence and large time behavior of solutions to Cauchy problem for a P1-approximation radiation hydrodynamics model. The global-in-time existence result is established in the small perturbation framework around a stable radiative equilibrium states in Sobolev space $H^4({\mathbb{R}}^3)$. Moreover, when the initial perturbation is also bounded in $L^1({\mathbb{R}}^3)$, the $L^2$-decay rates of the solution and its derivatives are achieved accordingly. The proofs are based on the Littlewood–Paley decomposition techniques and elaborate energy estimates in different frequency regimes.