gradient estimates in orlicz space for nonlinear elliptic equations abstract in this paper we generalize gradient estimates in l p space to orlicz space for weak solutions of elliptic equations of p laplacian type with small bmo coefficients in reifenberg flat domains our results improve the known results for such equations using a harmonic analysis free technique

micropolar meets newtonian the rayleigh benard problem abstract we consider the rayleigh benard problem for the two dimensional boussinesq system for the micropolar fluid our main goal is to compare the value of the critical rayleigh number and estimates of the nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical navier stokes system our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous navier stokes fluid in particular the critical rayleigh number for the micropolar model is larger than that for the navier stokes one and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction the estimate of the fractal dimension of the global attractor for the considered problem is better than that for the same problem for the navier stokes system besides we obtain also other results which relate the heat convection in the micropolar fluids to the newtonian ones among them relations between the nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the rayleigh benard attractor for the considered model

global regularity for the 2d magnetic benard fluid system with mixed partial viscosity abstract inspired by the previous works the main goal of this paper is to establish the global well posedness and conditional regularity of the incompressible magnetic benard fluid system with mixed partial viscosity in two dimensions as a result our results generalize and extend the previous results

discrete data assimilation in the lorenz and 2d navier stokes equations consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing in this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time this paper studies discrete data assimilation for the lorenz equations and the incompressible two dimensional navier stokes equations in both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution

uniform in time error estimates for the postprocessing galerkin method applied to a data assimilation algorithm we apply the postprocessing galerkin method to a recently introduced continuous data assimilation downscaling algorithm for obtaining a numerical approximation of the solution of the two dimensional navier stokes equations corresponding to given measurements from a coarse spatial mesh under suitable conditions on the relaxation nudging parameter the resolution of the coarse spatial mesh and the resolution of the numerical scheme we obtain uniform in time estimates for the error between the numerical approximation given by the postprocessing galerkin method and the reference solution corresponding to the measurements our results are valid for a large class of interpolant operators including low fourier modes and local averages over finite volume elements notably we use here the two dimensional navier stokes equations as a paradigm but our results apply equally to other evolution equations such as the boussinesq system of benard convection and other oceanic and atmospheric circulation models

optimal regularity estimates for parabolic p harmonic equations in this paper optimal regularity estimates for weak solutions of quasilinear parabolic equations of p laplacian type with small bmo coefficients are investigated our results improve the known results for such equations using a harmonic analysis free technique

observers for compressible navier stokes equation we consider a multidimensional model of a compressible fluid in a bounded domain we want to estimate the density and velocity of the fluid based on the observations for only velocity we build an

determining finite volume elements for the 2d navier stokes equations we consider the 2d navier stokes equations on a square with periodic boundary conditions dividing the square into n equal subsquares we show that if the asymptotic behavior of the average of solutions on these subsquares finite volume elements is known then the large time behavior of the solution itself is completely determined provided n is large enough we also establish a rigorous upper bound for n needed to determine the solutions to the navier stokes equation in terms of the physical parameters of the problem

data assimilation algorithm for 3d benard convection in porous media employing only temperature measurements abstract in this paper we propose a continuous data assimilation downscaling algorithm for the benard convection in porous media using only discrete spatial mesh measurements of the temperature in this algorithm we incorporate the observables as a feedback nudging term in the evolution equation of the temperature we show that under an appropriate choice of the nudging parameter and the size of the mesh and under the assumption that the observed data is error free the solution of the proposed algorithm converges at an exponential rate asymptotically in time to the unique exact unknown reference solution of the original system associated with the observed finite dimensional projection of temperature data moreover we note that in the case where the observational measurements are not error free one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

global well posedness for the 2d boussinesq system with anisotropic viscosity and without heat diffusion abstract we establish global existence and uniqueness theorems for the two dimensional non diffusive boussinesq system with anisotropic viscosity acting only in the horizontal direction which arises in ocean dynamics models global well posedness for this system was proven by danchin and paicu however an additional smoothness assumption on the initial density was needed to prove uniqueness they stated that it is not clear whether uniqueness holds without this additional assumption the present work resolves this question and we establish uniqueness without this additional assumption furthermore the proof provided here is more elementary we use only tools available in the standard theory of sobolev spaces and without resorting to para product calculus we use a new approach by defining an auxiliary stream function associated with the density analogous to the stream function associated with the vorticity in 2d incompressible euler equations then we adapt some of the ideas of yudovich for proving uniqueness for 2d euler equations

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

algebraic montgomery yang problem the non cyclic case montgomery yang problem predicts that every pseudofree circle action on the 5 dimensional sphere has at most 3 non free orbits using a certain one to one correspondence kollar formulated the algebraic version of the montgomery yang problem every projective surface s with quotient singularities such that the second betti number b2 s 1 has at most 3 singular points if its smooth locus s0 is simply connected we prove the conjecture under the assumption that s has at least one non cyclic singularity in the course of the proof we classify projective surfaces s with quotient singularities such that i b2 s 1 ii h 1 s 0 mathbb z 0 and iii s has 4 or more singular points not all cyclic and prove that all such surfaces have pi 1 s 0 cong mathfrak a 5 the icosahedral group

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

hardy inequalities in orlicz spaces we establish a sharp extension in the framework of orlicz spaces of the n dimensional hardy inequality involving functions defined on a domain g their gradients and the distance function from the boundary of g

data assimilation for the navier stokes α equations abstract the well posedness of the data assimilation problem for the navier stokes equations on a bounded three dimensional domain is investigated the data assimilation procedures under consideration are the adjoint method of variational data assimilation 4d var and the method of continuous data assimilation concerning the adjoint method the existence of optimal initial conditions with respect to an observation dependent cost functional is proven the optimizers are characterized by a first order necessary condition involving the adjoint linearized navier stokes equations and conditions for the uniqueness of the initial conditions are given well posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided

the time viscosity splitting method for the boussinesq problem abstract in this article we consider the time viscosity splitting method for the boussinesq problem by using this technique the considered problem is decoupled into two subproblems and each subproblem is solved more easily than the original one in the first step two linear elliptic problems with semi implicit schemes for the convection terms are solved the advantage of using such numerical technique is that a linear system with the constant coefficient matrices is obtained and then the computation becomes easy in the second step a system consists of a stokes problem and a linear elliptic problem is solved the main results of this article include that 1 establish the stability of numerical solutions in the time viscosity splitting method 2 provide the convergence results of strongly second order for the velocity and temperature and strongly first order for the pressure finally some numerical results are provided to display the performances of the developed numerical algorithms

on the regularity for the boussinesq equations in a bounded domain we address the regularity for the 2d boussinesq equations with zero diffusivity and positive viscosity on an open bounded smooth domain in particular we prove that for the initial velocity in h2 and the initial density in h1 there exists a global in time solution for which the velocity and the density are locally in time bounded functions in h2 and h1 respectively

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

continuous data assimilation for the 2d benard convection through velocity measurements alone 2015 elsevier b v an algorithm for continuous data assimilation for the two dimensional benard convection problem is introduced and analyzed it is inspired by the data assimilation algorithm developed for the navier stokes equations which allows for the implementation of variety of observables low fourier modes nodal values finite volume averages and finite elements the novelty here is that the observed data is obtained for the velocity field alone i e no temperature measurements are needed for this algorithm we provide conditions on the spatial resolution of the observed data under the assumption that the observed data is free of noise which are sufficient to show that the solution of the algorithm approaches at an exponential rate the unique exact unknown solution of the benard convection problem associated with the observed finite dimensional projection of velocity

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

global in time stability and accuracy of imex fem data assimilation schemes for navier stokes equations abstract we study numerical schemes for incompressible navier stokes equations using imex temporal discretizations finite element spatial discretizations and equipped with continuous data assimilation a technique recently developed by azouani et al 2014 we analyze stability and accuracy of the proposed methods and are able to prove well posedness long time stability and long time accuracy estimates under restrictions of the time step size and data assimilation parameter we give results for several numerical tests that illustrate the theory and show that for good results the choice of discretization parameter and element choices can be critical