localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3d nse the first part of the paper provides spatio temporal localization of a family of analytic regularity classes for the 3d nse obtained by beirao da veiga space time integrability of the gradient of the velocity on mathbb r 3 times 0 t which is out of the range of the sobolev embedding theorem reduction to the classical foias ladyzhenskaya prodi serrin space time integrability conditions on the velocity as well as the localization of the beale kato majda regularity criterion time integrability of the l norm of the vorticity the second part introduces a family of local scaling invariant hybrid geometric analytic classes in which coherence of the vorticity direction serves as a weight in the local spatio temporal integrability of the vorticity magnitude

finite element methods for maxwell s transmission eigenvalues the transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory and more importantly are parts of inverse algorithms for estimating material properties in this paper we propose two finite element methods to compute a few lowest maxwell s transmission eigenvalues which are of interest in applications since the discrete matrix eigenvalue problem is large sparse and in particular non hermitian due to the fact that the problem is neither elliptic nor self adjoint we devise an adaptive method which combines the arnoldi iteration and estimation of transmission eigenvalues exact transmission eigenvalues for balls are derived and used as a benchmark numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues

generic solvability of the axisymmetric 3 d euler equations and the 2 d boussinesq equations abstract in this paper we first prove a new criterion of global solvability for smooth axisymmetric solutions of the 3 d euler equations for a cylindrical type of domain which can be whole r3 then using this result we obtain generic solvability of the equations in the cylindrical domain which does not include the axis of symmetry by a similar method we also obtain generic solvability of the 2 d boussinesq system

a two grid discretization scheme of non conforming finite elements for transmission eigenvalues abstract in this paper for the helmholtz transmission eigenvalue problem we propose a two grid discretization scheme of non conforming finite elements with this scheme the solution of the transmission eigenvalue problem on a fine grid h is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid h and the solutions of two linear algebraic systems with the same positive definite hermitian and block diagonal coefficient matrix on the fine grid h we prove the resulting solution still maintains an asymptotically optimal accuracy and we report some numerical examples in two dimension and three dimension on the modified zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues

a new class of zienkiewicz type non conforming element in any dimensions in this paper a new class of zienkiewicz type non conforming finite element in n spatial dimensions with n 2 is proposed the new finite element is proved to be convergent for the biharmonic equation

a new multigrid finite element method for the transmission eigenvalue problems numerical methods for the transmission eigenvalue problems are hot topics in recent years based on the work of lin and xie 2015 we build a multigrid method to solve the problems with our method we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value problems on the finer and finer meshes theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues

a type of adaptive c0 non conforming finite element method for the helmholtz transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering in this paper for the helmholtz transmission eigenvalue problem in r d d 2 3 we discuss the a posteriori error estimates and adaptive algorithm of a type of c 0 non conforming finite elements including the morley zienkiewicz element and the modified zienkiewicz element we give the a posteriori error estimators for approximate eigenpair and prove their reliability and efficiency for eigenfunctions we also analyze their reliability for eigenvalues numerical results not only in r 2 but also in r 3 indicate that the a posteriori error estimators are sharp and the adaptive algorithm is efficient for computing real and complex transmission eigenvalues as expected

non conforming finite element methods for transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory in this paper for the helmholtz transmission eigenvalue problem we give a weak formulation which is a nonselfadjoint linear eigenvalue problem based on the weak formulation we first discuss the non conforming finite element approximation and prove the error estimates of the discrete eigenvalues obtained by the adini element the morley zienkiewicz element the modified zienkiewicz element et al and we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem

long time behavior of the two dimensional boussinesq equations without buoyancy diffusion abstract we study the global well posedness and stability instability of perturbations near a special type of hydrostatic equilibrium associated with the 2d boussinesq equations without buoyancy a k a thermal diffusion on a bounded domain subject to stress free boundary conditions the boundary of the domain is not necessarily smooth and may have corners such as in the case of rectangles we achieve three goals first we establish the global in time existence and uniqueness of large amplitude classical solutions efforts are made to reduce the regularity assumptions on the initial data second we obtain the large time asymptotics of the full nonlinear perturbation in particular we show that the kinetic energy and the first order derivatives of the velocity field converge to zero as time goes to infinity regardless of the magnitude of the initial data and the flow stratifies in the vertical direction in a weak topology third we prove the linear stability of the hydrostatic equilibrium t y satisfying t y 0 and the linear instability of periodic perturbations when t y 0 numerical simulations are supplemented to corroborate the analytical results and predict some phenomena that are not proved the authors are pleased to dedicate this paper to professor edriss saleh titi on the occasion of his 60th birthday professor titi s myriad research contributions including contributions to the problem constituting the focus of this work and leadership in the mathematical fluid dynamics community serve as an inspiration to his students collaborators and colleagues

on the blow up of solutions of the 3 d euler equations in a bounded domain it is shown that if 0 hat t is the maximal interval of existence of a smooth solutionu of the incompressible euler equations in a bounded simply connected domain subseteq r3 then int 0 hat t left omega cdot t right l infty omega dt infty where u is the vorticity crucial to this result is a special estimate proven in of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity

mixed methods for the helmholtz transmission eigenvalues the ciarlet raviart mixed finite element method is popular for the biharmonic equations in this paper we apply this method to the helmholtz transmission eigenvalue problem which is quadratic and

a two grid discretization scheme for eigenvalue problems a two grid discretization scheme is proposed for solving eigenvalue problems including both partial differential equations and integral equations with this new scheme the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy

regularity criterion of axisymmetric weak solutions to the 3d navier stokes equations abstract we consider the regularity of axisymmetric weak solutions to the navier stokes equations in r 3 let u be an axisymmetric weak solution in r 3 0 t w curl u and w be the azimuthal component of w in the cylindrical coordinates chae lee d chae j lee on the regularity of axisymmetric solutions of the navier stokes equations math z 239 2002 645 671 proved the regularity of weak solutions under the condition w l q 0 t l r with 3 2 r 2 q 3 r 2 we deal with the marginal case r which they excluded it is proved that u becomes a regular solution if w l 1 0 t b 0

on the blowup criteria and global regularity for the non diffusive boussinesq equations with temperature dependent viscosity coefficient abstract in this paper we consider the boussinesq equations with temperature dependent viscosity and zero thermal diffusivity a blowup criterion of classical solution depending only on l q 0 t l p with p 2 and 1 p 1 q 1 2 is obtained for the two dimensional equations this is in particular consistent with the results in chae 2006 hou and li 2005 lai et al 2011 where the global regularity of the 2d non diffusive boussinesq equations with constant viscosity coefficient i e const was proved a serrin s type blowup criterion depending on u l s 0 t l r l q 0 t l p with r p 3 3 r 2 s 1 and 3 p 2 q 1 is obtained for the three dimensional case we also study the global regularity with large data for the two dimensional equations in a non divergence form

on the volume growth of alexandrov spaces with nonnegative curvature abstract recently zhang and zhu introduced a new definition for ricci curvature bounded below on alexandrov spaces in this paper we give a volume growth lower bound estimate of alexandrov spaces with nonnegative ricci curvature in the sense of zhang zhu which extends yau s result from riemannian manifolds to alexandrov spaces

remarks on the regularity criteria of generalized mhd and navier stokes systems we study the regularity criteria of the three dimensional generalized magnetohydrodynamics mhd and navier stokes systems in particular we show that the regularity criteria of the generalized mhd system may be reduced to depend only on two diagonal entries of the jacobian matrix of the velocity vector field or one vorticity component and one entry of the jacobian matrix of the velocity vector field

criteria for the regularity of the solutions to the navier stokes equations based on the velocity gradient abstract we study the regularity of solutions to the navier stokes equations in the whole three dimensional space under the assumption that some additional conditions are imposed on one or more entries of the velocity gradient many such results with conditions using the lebesgue spaces can be found in the literature starting with the classical beirao da veiga s result the main goal of the present paper is to generalize the known results by replacing the standard lebesgue spaces by wider besov spaces in space variables in most such cases the technique of the proof leads to the deterioration of the prodi serrin scale so the results with the besov spaces are not still quite satisfactory nevertheless our technique leads to extensions and improvements of some results from the literature

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

regularity criterion of weak solutions to the 3d navier stokes equations via two velocity components abstract this paper is concerned with the regularity criterion of leray hopf weak solutions to the 3d navier stokes equations with respect to serrin type condition on two velocity filed components it is shown that the weak solution u u 1 u 2 u 3 is regular on 0 t if there exist two solution components for example u 2 and u 3 satisfying the condition u 2 u 3 l p 1 0 t l q 1 r 3 for 2 p 1 3 q 1 2 3 2 q 1

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

the shifted inverse iteration based on the multigrid discretizations for eigenvalue problems the shifted inverse iteration based on the multigrid discretizations developed in recent years is an efficient computation method for eigenvalue problems in this paper for general self adjoint ei

boundary layer effects for the 2 d boussinesq equations with vanishing diffusivity limit in the half plane nsfc 40890150 10801111 10971171 national basic research program 2005cb321700 natural science foundation of fujian province of china 2010j05011 fundamental research funds for the central universities 2010121006

three scale finite element discretizations for quantum eigenvalue problems based on globally and locally coupled discretizations some three scale finite element schemes are proposed in this paper for a class of quantum eigenvalue problems it is shown that the solution of a quantum eigenvalue problem on a fine grid may be reduced to the solution of an eigenvalue problem on a relatively coarse grid and the solutions of linear algebraic systems on a globally mesoscopic grid and the locally fine grid and the resulting solution is still very satisfactory

spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem we first develop an efficient spectral galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem then we present an iterative scheme based on computation of the first transmission eigenvalue to estimate the index of refraction of an inhomogeneous medium we present ample numerical results to demonstrate the effectiveness and accuracy of our approach

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

navier stokes equations with regularity in one direction we consider sufficient conditions for the regularity of leray hopf solutions of the navier stokes equations we prove that if the third derivative of the velocity u x3 belongs to the space lts0lxr0 where 2 s0 3 r0 2 and 9 4 r0 3 then the solution is regular this extends a result of beirao da veiga chin ann math ser b 16 407 412 1995 c r acad sci ser i math 321 405 408 1995 by making a requirement only on one direction of the velocity instead of on the full gradient the derivative u x3 can be substituted with any directional derivative of u

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence