Refined a priori estimates for the axisymmetric Navier-Stokes equations

In this paper, we consider the axisymmetric Navier-Stokes equations, and provide a refined a priori estimate for the swirl component of the vorticity. This extends Theorem 2 of [D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645--671].     

A priori and universal estimates for global solutions of superlinear degenerate parabolic equations

We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion equation u _t =Δu ^m +u ^p in a bounded domain with zero Dirichlet boundary conditions. Received: October 1, 2001?Published online: July 9, 2002     

A priori estimates for two multiscale finite element methods using multiple global fields to wave equations

We consider a scalar wave equation with nonseparable spatial scales. If the solution of the wave equation smoothly depends on some global fields, then we can utilize the global fields to construct multiscale finite element basis functions. We present two finite element approaches using the global fields: partition of unity method and mixed multiscale finite element method. We derive a priori error estimates for the two approaches and theoretically investigate the relation between the smoothness of the global fields and convergence rates of the approximations for the wave equation. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

A priori and a posteriori estimates for three‐dimensional Stokes equations with nonstandard boundary conditions

In this article, we study the Stokes problem with some nonstandard boundary conditions. The variational formulation decouples into a system for the velocity and a Poisson equation for the pressure. The corresponding discrete system do not need an inf‐sup condition. Hence, the velocity is approximated with “ curl ” conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a priori and a posteriori estimates and we finally concluded with numerical tests. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Navier–Stokes equations with regularity in one directional derivative of the pressure

This paper concerns the 3D Navier‐Stokes equations and prove an almost Serrin‐type regularity criterion in terms of one directional derivative of the pressure. Copyright © 2014 John Wiley & Sons, Ltd.     

Ladder estimates for micropolar fluid equations and regularity of global attractor

This paper is devoted to obtain ladder inequalities for 2D micropolar fluid equations on a periodic domain Q=(0, L)². The ladder inequalities are differential inequalities that connect the evolution of L² norms of derivatives of order N with the evolution of the L² norms of derivatives of other (usually lower) order. Moreover, we find (with slight assumption on external fields) long‐time upper bounds on the L² norms of derivatives of every order, which implies that a global attractor is made up from C^∞ functions. Copyright © 2010 John Wiley & Sons, Ltd.     

Uniform a priori estimates for a class of horizontal minimal equations

In the product space we obtain uniform a priori C ⁰ horizontal length estimates, uniform a priori C ¹ boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive an uniform global a priori C ¹ estimates and we infer an existence result.     

Some a priori estimates of solutions to the Maxwell equations

In this paper, we present a collection of a priori estimates of the electromagnetic field scattered by a general bounded domain. The constitutive relations of the scatterer are in general anisotropic. Surface averages are investigated, and several results on the decay of these averages are presented. The norm of the exterior Calderón operator for a sphere is investigated and depicted as a function of the frequency. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence and asymptotic behavior for the compressible Navier–Stokes equations with a non‐autonomous external force and a heat source

In this paper, we prove the global existence and asymptotic behavior, as time tends to infinity, of solutions in Hⁱ (i=1, 2) to the initial boundary value problem of the compressible Navier–Stokes equations of one‐dimensional motion of a viscous heat‐conducting gas in a bounded region with a non‐autonomous external force and a heat source. Some new ideas and more delicate estimates are used to prove these results. Copyright © 2008 John Wiley & Sons, Ltd.     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001     

On the regularity criterion for the solutions of 3D Navier–Stokes equations in weak multiplier spaces

In this work, we improved the regularity criterion on the Cauchy problem for the Navier–Stokes equations in multiplier space in terms of the two partial derivatives of velocity fields, ?₁u₁ and ?₂u₂. Copyright © 2011 John Wiley & Sons, Ltd.     

Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.     

Dini estimates for nonlocal fully nonlinear elliptic equations

We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order $\sigma \in (0,2)$ with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined $C^{\sigma +\alpha }$ estimate in [9].     

The limit behavior of the second‐grade fluid equations

In the paper, the limit behavior of solutions to the second‐grade fluid system with no‐slip boundary conditions is studied as both ν→0 and α→0. More precisely, it is verified that the convergence from second‐grade fluid system to Euler system holds as ν→0 and α→0 independently under the radial symmetry case.     

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

In this article, we investigate the L^∞(L²) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

一类广义长短波方程组在无界区域上的整体吸引子

研究了一类广义长短波方程组在无界区域上的解的长时间行为.通过引进加权空间,应用内插不等式和在加权空间的先验估计,证明了长短波方程组整体吸引子的存在性.     

A priori estimates of the solution for the Dirichlet problem

We study the Dirichlet problem defined by u = f in and u =g in when is the half-space or the unitary rectangle, obtainingan a priori estimate of the solution. Furthermore, in both casesa concrete numerical estimation is arrived at. First we getthe a priori estimate in the case of the half-space. The problemis resolved for the rectangle by initially translating it intothe half-space and using the results we had obtained for it,so that the problem can then be reduced back to the rectangle.Because of this we first establish an extension of the Sobolevspace of second order in the rectangle to the one defined inthe half-space.     

A priori error estimates for elliptic optimal control problems with a bilinear state equation

In this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered. The control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints. We derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples.     

Existence and asymptotic behavior of global smooth solution for p‐system with nonlinear damping and fixed boundary effect

This paper is concerned with the initial boundary value problem for the p‐system with nonlinear damping and fixed boundary condition. We show that the corresponding problem admits a unique global solution, and such a solution tends time asymptotically to the corresponding nonlinear diffusion wave governed by the classical Darcy's law provided that the corresponding prescribed initial error function is sufficiently small. Copyright © 2013 John Wiley & Sons, Ltd.