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In this note, we prove a blow-up criterion for the density-dependent incompressible Euler system in a bounded domain.

We study the Cauchy problem and vanishing dispersion limit of the Schrodinger-improved Boussinesq equations in R n .

In this paper, we prove a regularity criterion ψ , A ∈ W : = { ( ψ , A ) | ∇ ψ ∈ L 2 ( 0 , T ; L n ) , A ∈ L ∞ ( 0 , T ; L n ) ∩ L 2 p p − n ( 0 , T ; L p ) } with n p ≤ ∞ , for the nD ( n ≥ 3 ) time-dependent Ginzburg-Landau model in superconductivity. Finally, we show that this criterion holds true when n = 3 or 4.

In this paper, we establish some regularity criteria for both the simplified and the general Ericksen–Leslie system in R 3 .

In this paper we prove the global-in-time existence of smooth solutions of the 2D generalized magnetohydrodynamic equations with zero viscosity, when the power of the Laplacian operator −Δ in the dissipation term of the induction equation is between 1 and 2. We also show a regularity criterion on the direction of the magnetic field b 0 : = b | b | when β = 1 .

In this paper, we prove the global-in-time and uniform-in-ϵ existence of the 1-d complete equations for an electromagnetic fluid in a bounded interval without vacuum, where ϵ is the dielectric constant. Consequently, the limit as ϵ → 0 can be established. This approximation is usually referred to as the magnetohydrodynamic approximation, and is equivalent to the neglect of the displacement current.

We prove a global-in-time regularity which is uniform in the vanishing viscosity limit for the 2D Boussinesq system in a domain Ω : = ( − ∞ , ∞ ) × ( 0 , 1 ) with a slip boundary condition.

In this paper, we investigate a hyperbolic–parabolic system arising from chemotaxis. A blow up criterion of local strong solutions to the model in R d is established based on the scaling invariance of the system. As a by-product, we prove a global-in-time well-posedness result when d = 1 for large initial data.

This note proves a regularity criterion ∇ b ∈ L 1 ( 0 , T ; BMO ( R 2 ) ) for the 2D MHD system with zero magnetic diffusivity.

We consider an inverse problem of determining a viscosity coefficient in the Navier–Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.

In this paper, we study regularity criteria for the Navier–Stokes–Landau–Lifshitz system. Using delicate estimates, the regularity criteria for smooth solution of Navier–Stokes–Landau–Lifshitz system in Besov spaces and the multiplier spaces are obtained. The Navier–Stokes–Landau–Lifshitz system is coupled system of the Navier–Stokes equation and Landau–Lifshitz system, our results generalize the related results for Navier–Stokes equation and Landau–Lifshitz system to our system.

This paper proves the uniqueness of solutions to the time dependent drift diffusion semiconductor equations in three dimensions with L 2 initial data. This answers the open problem raised by da Veiga.

This paper studies the Cauchy problem of the Euler–Landau–Lifshitz system arising in the nematic liquid crystal flows. We prove two logarithmically improved regularity criteria for local smooth solutions in the homogeneous Besov space.