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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.

We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.

We consider the regularity problem under the critical condition to the biharmonic map heat flow from ℜ4 to a smooth compact Riemannian manifold without boundary. Using Gagliardo-Nirenberg inequalities and delicate estimates, the Serrin type regularity criterion for the smooth solutions of biharmonic map heat flow is obtained without assuming a smallness condition on the initial energy. Our result improved the results of Lamm in 5 and 6 and generalized the results of Chang, Wang, Yang 1, Strzelecki 11 and Wang 13, 14 to non-stationary case.

In this paper, various logarithmically improved regularity criteria for the 3D MHD equations are established in terms of the velocity field and pressure respectively. These regularity criteria can be regarded as log in time improvements of the standard Serrin's criteria established before.

We prove that a smooth solution of the 3D 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 the bmo-norm.

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

We consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion ∇d,∂td∈L1(0,T;B˙∞,∞0(ℝ2)) for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.

This work studies a fluid system with the linear Soret effect. We prove a regularity criterion in the homogeneous Besov space B ∞ , ∞ 0 .

In this paper, we consider the Cauchy problem for a model of electro-hydrodynamics in Rd(d=2,3). When d=2, we prove that the model has a unique global-in-time smooth solution. When d=3, we prove a regularity criterion in terms of the vorticity field only.

The low Mach number limit for classical solutions of the compressible magnetohydrodynamic equations without thermal conductivity is, here, studied. A uniform existence result for the Cauchy problem in is proved under the assumption that the initial data are uniformly bounded with respect to the Mach number in and are well-prepared in . Copyright © 2011 John Wiley & Sons, Ltd.

We prove uniform local-in-time existence and uniqueness of solutions to the density-dependent magnetohydrodynamic equations.

We prove the uniqueness for weak solutions of the time-dependent 2-D Ginzburg-Landau model for superconductivity with L 2 initial data in the case of Coulomb gauge. This question was left open in Tang and Wang (Physica D, 88:139–166, 1995). We also prove the uniqueness of the 3-D radially symmetric solution in bounded annular domain with the choice of Lorentz gauge and L 2 initial data.

We study the compressible Navier-Stokes equations of viscous heat-conductive fluids in a periodic domain $\mathbb{T}^{3}$ with zero heat conductivity k=0. We prove a blow-up criterion for the local strong solutions in terms of the temperature and positive density, similar to the Beale-Kato-Majda criterion for ideal incompressible flows.

In this work, a regularity criterion is proved for local strong solutions of the Navier–Stokes equations in the presence of mass diffusion.

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

The local-in-time existence and uniqueness of strong solutions to the Euler equations in the whole space with nondecaying and certainly regular initial velocity are concerned. It is obtained that the spatial regularity of solutions coincides with that of initial velocity under the suitable setting of external forcing terms. Regularity criteria focusing into the vorticity are also discussed due to the similar arguments of Beale–Kato–Majda.

The global Cauchy problem for a 2D Leray- α -MHD model with zero viscosity is studied. The vanishing limit on α is also considered in this paper.

In this paper, we study the n -dimensional Lagrangian-averaged magnetohydrodynamic- α (LAMHD- α ) model with 5 ≤ n ≤ 7 throughout the whole space. Various regularity criteria are established for both the cases with and those without viscosity.

In this paper, the Cauchy problem for a Leray- α -MHD model is considered for n = 2 , 3 . When n = 2 , global existence for the strong solution is proved even for the cases with a partial viscous term. For the three-dimensional case, various regularity criteria are established.

This paper studies the Cauchy problem of the MHD equations with mass diffusion. We use the Tikhonov fixed point theorem to prove a local-in-time well-posedness theorem. Copyright © 2010 John Wiley & Sons, Ltd.

We study the $n$-dimensional magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model in the whole space. Various regularity criteria are established. When $n=4$, uniqueness of weak solutions is also proved. As a corollary, the strong solution to this model exists globally, as $n \leq 4$.

In this paper, logarithmically improved regularity criteria for the Navier–Stokes and the MHD equations are established in terms of both the vorticity field and the pressure.

We prove the uniqueness of weak solutions of the 3-D time-dependent Ginzburg-Landau equations for super-conductivity with initial data (ψ0, A0)∈ L2 under the hypothesis that (ψ, A) ∈ Ls(0, T; Lr,∞) ×(0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here Lr,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ0 ∈ , A0 ∈ L3 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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 prove the local well-posedness of strong solutions to a compressible non-isothermal model for nematic liquid crystals in a bounded domain Ω⊂R3 provided that the initial data satisfy a natural compatibility condition. The assumption on the positivity of initial density is not needed here, and it may vanish in an open subset of Ω.

We study the Cauchy problem of certain Boussinesq-α equations in n dimensions with n = 2 or 3. We establish regularity for the solution under ▽u ∈ L1 (0, T; Ḃ0∞,∞(ℝn)). As a corollary, the smooth solution of the Leray-α–Boussinesq system exists globally, when n = 2. For the Lagrangian averaged Boussinesq equations, a regularity criterion ▽θ ∈ L1(0, T;L∞(ℝ2)) is established. Other Boussinesq systems with partial viscosity are also discussed in the paper.

In this paper we prove the existence, uniqueness and stability of solutions of an inverse problem to a time-dependent Ginzburg--Landau model for superconductivity with the final overdetermination.

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.

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.

In this paper, various regularity criteria for the strong solutions to a problem arising in the study of magneto-elastic interactions are established. In particular, these regularity criteria are also true for the Landau-Lifshitz equation and give extensions of previous results.

We prove local in time existence and uniqueness of solutions of the ideal inhomogeneous magnetohydrodynamic equations.

We prove a regularity criterion of the strong solutions for a Bona–Colin–Lannes system in Besov space. As a byproduct, we show the existence of globally smooth solutions for a symmetric Bona–Colin–Lannes system.

We study the isentropic compressible Navier-Stokes equations with radially symmetric data and non-negative initial density in an annular domain. We prove the global existence of strong solutions for any $\gamma\geq 1$. Moreover, we obtain the uniform in time $L^{\infty}$-boundedness of the density and $H^{1}$-boundedness of the velocity, improving therefore the corresponding result in [2], where the condition $\gamma\geq 2$ is required to guarantee the existence.

In this paper, we will prove global well-posedness for the Cauchy problems of two modified-Leray-α-MHD models with partial viscous terms. Copyright © 2009 John Wiley & Sons, Ltd.

In this note we prove a regularity criterion ω ≔ curl u ∈ L 1 ( 0 , T ; B ∞ , ∞ 0 ) for the 3D Boussinesq system with partial viscosity. Here u is the velocity, ω is the vorticity and B ∞ , ∞ 0 denotes the homogeneous Besov space.

We use an interpolation inequality on Besov spaces to show some logarithmically improved regularity criteria for Navier–Stokes equations, the harmonic heat flow, the Landau–Lifshitz equations, and the Landau–Lifshitz–Maxwell system. Copyright © 2009 John Wiley & Sons, Ltd.

In this note we prove the uniqueness of weak solutions to a nonlinear hyperbolic system in electrohydrodynamics without the effects of a dissociation–recombination process. It is still open in the presence of a special dissociation–recombination process, although the existence of at least one weak solution was proved via the method of renormalized solutions by Feireisl [E. Feireisl, Weak solutions to a non-linear hyperbolic system arising in the theory of dielectric liquids, Math. Methods Appl. Sci. 18 (1995) 1041–1052] in 1995.

We consider the 3D density-dependent Boussinesq equations and the classical Boussinesq equations with partial viscosity terms. We prove some regularity criteria of strong solutions for the Boussinesq equations.

We consider the compressible magnetohydrodynamic (MHD) equations with nonnegative thermal conductivity or infinite electric conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set.

We obtain logarithmic improvements for conditions of regularity in the 3D Navier-Stokes equations.

We consider the hydrodynamic theory of liquid crystals. We prove some regularity criteria for a simplified Ericksen-Leslie system. The existence and uniqueness of global smooth solutions is also proved for a regularization model of this simplified system.

We prove some regularity conditions for the MHD equations with partial viscous terms and the Leray-$\alpha$-MHD model. Since the solutions to the Leray-$\alpha$-MHD model are smoother than that of the original MHD equations, we are able to obtain better regularity conditions in terms of the magnetic field $B$ only.

The global Cauchy problem for the 2-D magnetohydrodynamic-α models with partial viscous terms is studied. The vanishing limit on α is also considered in this paper.

We prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic (MHD) equations with the coefficients depending on the temperature. As a by-product, we give a simple proof for the nonexistence of nontrivial weak time-periodic solutions by the entropy principle of Clausius–Duhem and a new Poincare-type inequality.

In this article, we prove the local solvability of an inverse problem to the density-dependent Navier–Stokes equations in the case of integral overdetermination.

We consider the regularity problem under the critical condition to some liquid crystal models and the Landau-Lifshitz equations. The Serrin type reularity criteria are obtained in the terms of the Besov spaces.

We prove the asymptotic stability for weak solutions to the 3-D Navier-Stokes equations in the class \(\nabla u \in L^{1}(0, \infty; \dot{B}^{0}_{\infty,\infty}({\mathbb{R}}^{3})) \cap LLogL(0, \infty; \dot{B}^{0}_{\infty,\infty}({\mathbb{R}}^{3}))\) with arbitrary initial and external perturbations. This solves a problem due to Yong Zhou (Proc. Roy. Soc. Edinburgh, 136A (2006), 1099-1109).

We study the Navier–Stokes equations of three-dimensional compressible isentropic and two-dimensional heat-conducting flows in a domain Ω with nonnegative density, which may vanish in an open subset (vacuum) of Ω, and with positive density, respectively. We prove some blow-up criteria for the local strong solutions.

We study an initial boundary value problem for the equations of plane magnetohydrodynamic compressible flows, and prove that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. As a by-product, this paper improves the related results obtained by Frid and Shelukhin for the case when the magnetic effect is neglected.

We prove the Lipschitz continuous dependence on initial data of global spherically symmetric weak solutions to the Navier–Stokes equations of a viscous polytropic ideal gas in bounded annular domains with the initial data in the Lebesgue spaces. Copyright © 2007 John Wiley & Sons, Ltd.