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This paper poses a new question and proves a related result. Particularly, the nonexistence of a nontrivial time-periodic solution to the Navier–Stokes system is proved in a bounded domain in $ \mathbb{R}^2 $.

In this paper, we provide a blow-up criterion for the density-dependent incompressible magnetohydrodynamic system with zero viscosity. The proof uses the Lp-method and the Kato–Ponce inequalities in the harmonic analysis. The novelty of our work lies in the fact that we deal with the case in which the resistivity η is positive.

First, we prove uniform-in-$ \epsilon $ regularity estimates of local strong solutions to the Chern-Simons-Schrödinger equations in $ \mathbb{R}^2 $. Here $ \epsilon $ is the dispersion coefficient. Then we prove the global well-posedness of strong solutions to the limit problem $ (\epsilon = 0) $.

In this paper, we will use the Banach fixed point theorem to prove the uniform-in-ϵ existence of the compressible full magneto-micropolar system in a bounded smooth domain, where ϵ is the dielectric constant. Consequently, the limit as ϵ→0 can be established. This approximation is usually referred to as the magnetohydrodynamics approximation and is equivalent to the neglect of the displacement current.

In this paper, we are concerned with the uniform regularity estimates of smooth solutions to the isentropic compressible magneto-micropolar system in . Under the assumption that , and by applying the classic bilinear commutator and product estimates, the uniform estimates of solutions to the isentropic compressible magneto-micropolar system are established in space, .

We prove the existence of a unique solution for a time-dependent Ginzburg-Landau model in superconductivity under the Coulomb gauge. Also we prove the uniform-in-$\epsilon$ well-posedness of the solution, where $\epsilon$ is the coefficient of the double-well potential energy.

In this paper, we consider the n(n≥3)dimensional time-dependent Ginzburg–Landau model for superconductivity. First, we obtain a global existence of very weak solutions. Finally we prove a result of weak-very weak uniqueness.

We prove uniform-in-$\mu$ estimates for a mathematical model in superfluidity. Consequently, the limit as $\mu\to0$ can be established.

In this article we prove the local well-posedness for an Ericksen-Leslie's parabolic-hyperbolic compressible non-isothermal model for nematic liquid crystals with positive initial density.

Abstract We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω.

In this paper, we first establish a regularity criterion for the strong solutions to the density-dependent incompressible MHD system with zero resistivity in a bounded domain. Then we use it and the bootstrap argument to prove the global well-posedness provided that the initial data u0 and b0 satisfy that (d-2)||∇u0 || L2+||b0||w1,p are sufficiently small with . We do not assume the positivity of initial density, it may vanish in an open subset (vacuum) of Ω.

We obtain some regularity criteria for the wave map, a liquid crystals model, and the Hall-MHD with ion-slip effect.

Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them emph{manifolds of discontinuities}. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this paper, we gave a parabolic version of their results. That is, we gave a gradient estimate for parabolic equations of divergence forms with piecewise smooth coefficients. The coefficients are assumed to be independent of time and their discontinuities are likewise the previous elliptic equations. As an application of this estimate, we also gave a pointwise gradient estimate for the fundamental solution of a parabolic operator with piecewise smooth coefficients. Both gradient estimates are independent of the distances between manifolds of discontinuities.

We prove the existence of strong global solutions of the 1-D modified compressible Navier-Stokes-Fourier equations proposed by Howard Brenner [1,2].

We consider the regularity of weak solutions to the Navier-Stokes equations in $mathbb{R}^3$. Let $u:=(u_1,u_2,u_3)$ be a weak solution and $widetilde{u}:=(u_1,u_2,0)$. We prove that $u$ is strong solution if $ ablawidetilde{u}$ satisfy Serrin's type criterion.

In this paper, we prove that for a semilinear wave equation with source terms, the energy decays exponentially as time approaches infinity. For this end we use the the multiplier method.

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocity u0 and small ∇d0. We also give a regularity criterion ∇d∈Lp(0,T;Lq(Ω)) (2/q)+(2/p)=1, 2

This paper studies uniqueness of weak solutions to an electrohydrodynamics model in ℝd (d=2,3). When d=2, we prove a uniqueness without any condition on the velocity. For d=3, we prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov space.

We prove that a smooth solution of the 3D Cahn-Hilliard-Boussinesq system with zero viscosity in a bounded domain breaks down if a certain norm of vorticity blows up at the same time. Here, this norm is weaker than bmo-norm.

We establish some regularity criteria for a turbulent magnetohydrodynamic model. As a corollary, we prove that the smooth solution exists globally when the spatial dimension n satisfies 3≤n≤8.

A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.

We prove a regularity criterion ∇À∈L2/3(0,T;BMO) for weak solutions to the Navier-Stokes equations in three-space dimensions. This improves the available result with L2/3(0,T;L∞).