A Regularity Criterion for the Navier–Stokes Equations

We prove that a weak solution u = (u ¹, u ², u ³) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrin's criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L ^{6, ∞}.     

First-order differentiability of the flow of a system withL p controls

This paper considers the study of the regularity of the flow of a nonautonomous nonlinear control process when the set of control maps is endowed with theL ^p -topology. Roughly speaking, it is proved that, if the norm of the mapf(t, x, u) defining the process together with its first derivatives goes to infinity, with the norm ofu not faster thanu ^p ,p>1, then the flow isC ¹ in theL ^p -topology. This property implies that, if the control maps are bounded, then the flow is differentiable in anyL ^p ,p>1. Moreover, it is proved that the only systems for which the flow is differentiable inL ¹ are the affine ones.This research was supported by a grant from Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica, Rome, Italy.     

Extra regularity for parabolic equations with drift terms

We consider a parabolic equation with a drift term u+bu–u_t=0. Under some natural conditions on the vector valued function b, we prove that solutions possess extra regularity and better qualitative behavior than those provided by standard theory. For example, we show that the fundamental solution has global Gaussian upper bound even for some b with a large singularity in the form of c/|x|. We also show that bounded solutions are Hölder continuous when |b|² is just form bounded and divergence free, a case where not even continuity is expected. A Liouville type theorem is also proven.     

On the Blow-up Criterion of Smooth Solutions to the MHD System in BMO Space

In this paper we study the blow-up criterion of smooth solutions to the incompressible magnetohydrodynamics system in BMO space. Let （u（x,t）,b（x,t）） be smooth solutions in （0, T）. It is shown that the solution （u（x, t）, b（x, t）） can be extended beyond t = T if （u（x,t）, b（x, t）） ∈ L^1 （0, T; BMO） or the vorticity （rot u（x, t）, rot b（x, t）） ∈ L^1 （0, T; BMO） or the deformation （Def u（x, t）, Def b（x, t）） ∈ L^1 （0, T; BMO）.     

Some solutions for the spin glass system and for systems originating from it

For the spin glass system the solutions, u = νΦ and u = νx/|x| are considered, where ν is a scalar function and Φε?³ is constant. For a system originating from the spin glass one, a solution u = tan[q(|x|/t)]x/|x| is considered.     

Partial regularity criteria for suitable weak solutions of the three‐dimensional liquid crystals flow

In this paper, we establish some new interior regularity criteria for suitable weak solutions of the liquid crystals flow in terms of the smallness of the scaled L^p,q‐norm of the velocity field or the vorticity, which extends the results by Scheffer in [Communications in Mathematical Physics 1980; 73 :1–42]. Copyright © 2016 John Wiley & Sons, Ltd.     

Global Regularity of the 3D Axi-Symmetric Navier–Stokes Equations with Anisotropic Data

In this paper, we study the 3D axisymmetric Navier–Stokes equations with swirl. We prove the global regularity of the 3D Navier–Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some L ^p norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.     

Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data

The method introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity (L ^p , Marcinkiewicz or C ^0,α ) of the weak solutions of Dirichlet problems hinges on the handle of inequalities concerning the integral of on the subsets where |u(x)| is greater than k. In this framework, here we give a contribution with the study of the Marcinkiewicz regularity of the gradient of infinite energy solutions of Dirichlet problems with nonregular data. Dedicated to Juan Luis Vazquez for his 60th birthday (“El verano del Patriarca”, see [19]).     

Lorentz space estimates and Jacobian convergence for the Ginzburg-Landau energy with applied magnetic field

In this paper, we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in [15]. We focus on getting estimates for the Ginzburg-Landau energy with external magnetic field h _ex in certain interesting regimes of h _ex . This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration (u, A) is comparable to the L ^{2, ∞} Lorentz space norm of ∇ _A u. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined in terms of Lorentz spaces. Supported by an NSF Graduate Research Fellowship.     

A posteriori error estimates of stabilized finite volume method for the Stokes equations

In this work, the residual‐type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in H¹ norm and pressure in L² norm are derived. Furthermore, a global upper bound of u ? u_h in L²‐norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.     

Variational aspects of the Seiberg-Witten functional

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection A on a line bundleL and a section of another bundleW ⁺ constructed fromL and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

The existence,uniqueness, and regularity for an incompressible Newtonian flow with intrinsic degree of freedom

We consider the regularity and uniqueness of solution to the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid, taking into account internal degree of freedom. We first show there exist uniquely a local strong solution. Then we show this solution can be extend to the whole interval [0,T] if the velocity u, or its gradient ? u, or the pressure p belongs to some function class, which are similar with that of incompressible Navier–Stokes equations. Our result shows that the solution is unique in these classes, and that velocity field plays a more prominent role in the existence theory of strong solution than the angular velocity field. Finally, if the L^{3 ∕ 2}‐norm of the initial angular velocity vector and some homogeneous Besov norm of initial velocity field are small, then there exists uniquely a global strong solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations

In this paper we study the magneto-micropolar fluid equations in ℝ³, prove the existence of the strong solution with initial data in H^s(ℝ³) for , and set up its blow-up criterion. The tool we mainly use is Littlewood–Paley decomposition, by which we obtain a Beale–Kato–Majda-type blow-up criterion for smooth solution (u, ω, b) that relies on the vorticity of velocity ∇ × u only. Copyright © 2007 John Wiley & Sons, Ltd.     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).     

Error analysis of the L2 least‐squares finite element method for incompressible inviscid rotational flows

In this article we analyze the L² least‐squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity‐vorticity‐pressure formulation. The least‐squares functional is defined in terms of the sum of the squared L² norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first‐order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H¹ norm for velocity and pressure and a suboptimal rate of convergence in the L² norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x₀,t₀), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L^∞ norm as long as that either the scaled local L²-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

Averaging of directional derivatives in vertices of nonobtuse regular triangulations

For a shape-regular triangulation ${\mathcal{T}_h}For a shape-regular triangulation _h{\mathcal{T}_h} without obtuse angles of a bounded polygonal domain W ì ?²{\Omega\subset\Re^2} , let L_h{\mathcal L_h} be the space of continuous functions linear on the triangles from T_h{\mathcal{T}_h} and Π _h the interpolation operator from C([`(W)]){C(\overline\Omega)} to L<sub>h</sub>{\mathcal L_h} . This paper is devoted to the following classical problem: Find a second-order approximation of the derivative ?u/?z(a){\partial u/\partial z(a)} in a direction z of a function u ? C<sup>3</sup>([`(W)]){u\in C^3(\overline\Omega)} in a vertex a in the form of a linear combination of the constant directional derivatives ?P_h(u)/?z{\partial \Pi_h(u)/\partial z} on the triangles surrounding a. An effective procedure for such an approximation is presented, its error is proved to be of the size O(h ²), an operator W_h: L_h?L_h×L_h{\mbox{W}_h: \mathcal L_h\longrightarrow\mathcal L_h\times\mathcal L_h} relating a second-order approximation W _h [Π _h (u)] of ?u{\nabla u} to every u ? C³([`(W)]){u\in C^3(\overline\Omega)} is constructed and shown to be a so-called recovery operator. The accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically.