global solutions to the two dimensional quasi geostrophic equation with critical or super critical dissipation the two dimensional quasi geostrophic 2d qg equation with critical and super critical dissipation is studied in sobolev space h s 2 for critical case open image in new window existence of global large solutions in h s is proved for s open image in new window when open image in new window is small this generalizes and improves the results of constantin d cordoba and wu 4 for s 1 2 and the result of a cordoba and d cordoba 8 for s open image in new window for s 1 these solutions are also unique the improvement for pushing s down from 1 to open image in new window is somewhat surprising and unexpected for super critical case 0 open image in new window existence and uniqueness of global large solution in h s is proved when the product open image in new window is small for suitable s 2 2 p 1 and 0 1

global regularity for a class of 2d generalized tropical climate models abstract this paper establishes the global existence and regularity of solutions to a two dimensional 2d tropical climate model tcm with fractional dissipation the inviscid counterpart of this model was derived by frierson majda and pauluis 8 as a model for tropical geophysical flows this model reflects the interaction and coupling among the barotropic mode u the first baroclinic mode v of the velocity and the temperature the systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization the aim here is the global regularity with the least dissipation we prove two main results first the global regularity of the system with v and for 1 and 3 2 and second the global regularity of the system with v for 3 2 the proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal the key tools employed here include the maximal regularity for general fractional heat operators the littlewood paley decomposition and besov space techniques lower bounds involving fractional laplacian and simultaneous estimates of several coupled quantities

an extension of the beale kato majda criterion for the euler equations the beale kato majda criterion asserts that smooth solutions to the euler equations remain bounded past t as long as is finite ohgr being the vorticity we show how to replace this by a weaker statement on where j is a frequency localization around

local well posedness for the maxwell schrodinger equation time local well posedness for the maxwell schrodinger equation in the coulomb gauge is studied in sobolev spaces by the contraction mapping principle the lorentz gauge and the temporal gauge cases are also treated by the gauge transform

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

the existence and uniqueness of the local solution for a camassa holm type equation a shallow water equation of camassa holm type containing nonlinear dissipative effect is investigated using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself we establish the existence and uniqueness of its local solution in sobolev space h s r with s 32 meanwhile a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order sobolev space h s with 1

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

the facets of the polyhedral set determined by the gale hoffman inequalities the gale hoffman inequalities characterize feasible external flow in a capacitated network among these inequalities those that are redundant can be identified through a simple arc connectedness criterion

regularization methods for optimization problems with probabilistic constraints we analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides we develop two numerical methods with regularization for their numerical solution the methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p efficient points the algorithms yield an optimal solution for problems involving concave probability distributions for arbitrary distributions the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions the methods are compared numerically to two cutting plane methods

the existence of global strong and weak solutions for the novikov equation abstract the well posedness of the global strong and weak solutions for the novikov equation is investigated provided that initial value u 0 h s s 3 2 and satisfying a sign condition the existence and uniqueness of global strong solutions for the equation are shown to be valid in sobolev space the estimates in h q r space with 0 q 1 2 which are derived from the equation itself are developed to prove the existence and uniqueness of the global weak solutions

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

nonselfadjoint schrodinger operators with singular first order coefficients schrodinger operators of the form t i grad b x 2 a x grad q x in r m are considered where a b ate real vector valued functions and q is a scalar complex valued function it is shown that t is essentially quasi m accretive in l 2 r m if 1 x2223 1 a l 4 l div a l and re q 0 the proof is elementary

spatial modeling of regional variables abstract in this article accumulated sudden infant death syndrome sids data from 1974 1978 and 1979 1984 for the counties of north carolina are analyzed after a spatial exploratory data analysis markov random field models are fit to the data the spatial trend is meant to capture the large scale variation in the data and the variance and spatial dependence are meant to capture the small scale variation the trend could be a function of other explanatory variables or could simply be modeled as a function of spatial location both models are fit and compared the results give an excellent illustration of a phenomenon already well known in time series that autocorrelation in data can be due to an undiscovered explanatory variable indeed for 1974 1978 we confirm a dependence of sids rate on proportion of nonwhite babies born along with insignificant spatial correlation without this regressor variable however the spatial correlation is significant in 1979 1984 perhaps due to reporting bias o

on the existence of a feasible flow in a stochastic transportation network many transportation networks e g networks of cooperating power systems and hydrological networks involve a real valued demand function defined on the set of nodes and it is said to be feasible if there exists a flow such that at each node the sum of the incoming flow values is greater than or equal to the demand assigned to this node by the theorem of d gale and a hoffman a system of linear inequalities involving the demand and the arc capacity functions gives necessary and sufficient condition for the feasibility of the demand if the demands and or the arc capacities are random then an important problem is to find the probability that all these inequalities are satisfied this paper proposes a new method to eliminate all redundant inequalities for given lower and upper bounds of the demand function and finds sharp lower and upper bounds for the probability that a feasible flow exists the results can be used to support transportation network analysis and design

single commodity stochastic network design under probabilistic constraint with discrete random variables single commodity networks are considered where demands at the nodes are random the problem is to find minimum cost optimal built in capacities at the nodes and arcs subject to the constraint that all demands should be met on a prescribed probability level reliability constraint and some deterministic constraints should be satisfied the reliability constraint is formulated in terms of the gale hoffman feasibility inequalities but their number is reduced by elimination technique the concept of a p efficient point is used in a smart way to convert and then relax the problem into an lp the p efficient points are simultaneously generated with the solution of the lp the joint distribution of the demands is used to obtain the p efficient points for all stochastic inequalities that were not eliminated and the solution of a multiple choice knapsack problem is used to generate p efficient points the model can be applied to planning in interconnected power systems flood control networks design of shelter

well posedness travelling waves and geometrical aspects of generalizations of the camassa holm equation abstract in this paper we consider a five parameter equation including the camassa holm and the dullin gottwald holm equations among others we prove the existence and uniqueness of solutions of the cauchy problem using kato s approach conservation laws of the equation up to second order are also investigated from these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it the quadrature obtained is of capital importance in a classification of bounded travelling wave solutions we also find some explicit solutions given in terms of elliptic integrals finally we classify the members of the equation describing pseudo spherical surfaces

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

existence of a local smooth solution in probability to the stochastic euler equations in r3 abstract we establish the existence of a local smooth solution of the stochastic euler equations in r 3 probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data and the random noise there are numerous works on the stochastic euler equations in a two dimensional domain even for the deterministic euler equations in a three dimensional domain the only results are concerned with the local existence of smooth solutions our goal is to extend such local existence result to the stochastic equations

an improved regularity criterion for the 3d hall mhd equations via the vorticity abstract in this paper we consider the three dimensional incompressible hall magnetohydrodynamic equations and establish an improved regularity criterion of local in time classical solutions involving only the vorticity field consequently this improves the previous result

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

a minimal algorithm for the multiple choice knapsack problem abstract the multiple choice knapsack problem is defined as a 0 1 knapsack problem with the addition of disjoined multiple choice constraints as for other knapsack problems most of the computational effort in the solution of these problems is used for sorting and reduction but although o n algorithms which solve the linear multiple choice knapsack problem without sorting have been known for more than a decade such techniques have not been used in enumerative algorithms in this paper we present a simple o n partitioning algorithm for deriving the optimal linear solution and show how it may be incorporated in a dynamic programming algorithm such that a minimal number of classes are enumerated sorted and reduced computational experiments indicate that this approach leads to a very efficient algorithm which outperforms any known algorithm for the problem

a generalized linear models approach to spatial scan statistics for covariate adjustment the spatial scan statistic proposed by kulldorff commun statist theory methods 1997 26 1481 1496 is one of the most widely used methods for detecting spatial clusters and evaluating their statistical significance however it is not fully capable of adjusting for all types of confounding covariates in this article a generalized linear models glm approach to construct spatial scan statistics which is readily in a form for covariate adjustment is proposed using glm spatial scan statistics for different probability models can be formulated in a single framework the test statistic is based on the log likelihood ratio test statistic and evaluated using monte carlo hypothesis testing the proposed method is illustrated using texas female breast cancer data concerning late versus early stage cancer cases with covariates of race ethnicity and age group

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

on the dual systems to euler and navier stokes equations in r3 in this paper we consider two model systems of equations obtained by taking the hodge duals of the euler and the navier stokes systems respectively the vorticity formulation of the new system becomes a nonlinear non local scalar equation having similarities to the quasi geostrophic equations we prove the local existence of classical solutions and derive the beale kato majda type of blow up criterion for the dual euler system moreover we actually show that the finite time blow up occurs for generic non trivial initial data for this system for the dual navier stokes equations we construct an explicitly global in time smooth solution for general initial data using the hopf cole transform in the appendix we formulate the euler system and the dual euler system in terms of the clifford algebra

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

global unique solvability of nonhomogeneous asymmetric fluids a lagrangian approach abstract we show global existence and uniqueness of solutions for the 3d nonhomogeneous asymmetric fluids equations through a lagrangian approach in particular uniqueness of the solution is proved under quite soft assumptions about its regularity which brings the knowledge about nonhomogeneous asymmetric fluids to the same level as for the variable density navier stokes equations

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

log linear logistic model fitting and local score statistics for cluster detection with covariate adjustments the standard method for p value computation of spatial scan statistics with adjustments for covariate effects is to conduct monte carlo simulations with these effects estimated under the null hypothesis of no clustering however when the covariates are geographically unbalanced the proposed monte carlo p value estimates are too conservative with corresponding loss of power due to excessive adjustments for confounding between covariates and location we show that the use of an alternative procedure that involves local score statistics with parameters fitted on a log linear or logistic model addresses this problem we also discuss extensions of the procedure when there are multiple or continuous covariates

detection of spatial clustering with average likelihood ratio test statistics generalized likelihood ratio glr test statistics are often used in the detection of spatial clustering in case control and case population datasets to check for a significantly large proportion of cases within some scanning window the traditional spatial scan test statistic takes the supremum glr value over all windows whereas the average likelihood ratio alr test statistic that we consider here takes an average of the glr values numerical experiments in the literature and in this paper show that the alr test statistic has more power compared to the spatial scan statistic we develop in this paper accurate tail probability approximations of the alr test statistic that allow us to by pass computer intensive monte carlo procedures to estimate p values in models that adjust for covariates these monte carlo evaluations require an initial fitting of parameters that can result in very biased p value estimates

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

global well posedness to the cauchy problem of 2d density dependent micropolar equations with large initial data and vacuum abstract this paper studies the cauchy problem of the 2d nonhomogeneous incompressible micropolar equations it is shown that the problem admits a unique global strong solution note that the initial data can be arbitrary large and the initial density can contain vacuum states

vanishing viscosity for non homogeneous asymmetric fluids in r3 the l2 case abstract we study the vanishing viscosity problem for the local in time solutions to the equations of non homogeneous viscous incompressible asymmetric fluid in r 3 in the l 2 context we prove that the fluid variables converge uniformly as the viscosities go to zero to a solution of a non homogeneous non viscous incompressible asymmetric fluid governed by an euler like system this completes the previous work 5 where results for l p p 3 where obtained

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

global weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

global weak solutions for variable density asymmetric incompressible fluids abstract we establish the existence of global in time weak solutions for the equations of asymmetric incompressible fluids with variable density when the initial density is not necessarily strictly positive

small amplitude solutions of the generalized imbq equation in this paper the global existence of small amplitude solution for the cauchy problem of the multidimensional generalized imbq equation is proved moreover we obtain a nonlinear scattering result of the cauchy problem of the imbq equation for small initial data

well posedness in energy space for the periodic modified benjamin ono equation abstract we prove that the periodic modified benjamin ono equation is locally well posed in the energy space h 1 2 this ensures the global well posedness in the defocusing case the proof is based on an x s b analysis of the system after a gauge transform

mathematical programming approaches for generating p efficient points abstract probabilistically constrained problems in which the random variables are finitely distributed are non convex in general and hard to solve the p efficiency concept has been widely used to develop efficient methods to solve such problems those methods require the generation of p efficient points pleps and use an enumeration scheme to identify pleps in this paper we consider a random vector characterized by a finite set of scenarios and generate pleps by solving a mixed integer programming mip problem we solve this computationally challenging mip problem with a new mathematical programming framework it involves solving a series of increasingly tighter outer approximations and employs as algorithmic techniques a bundle preprocessing method strengthening valid inequalities and a fixing strategy the method is exact resp heuristic and ensures the generation of pleps resp quasi pleps if the fixing strategy is not resp is employed and it can be used to generate multiple pleps to the best of our knowledge generating a set of pleps using an optimization based approach and developing effective methods for the application of the p efficiency concept to the random variables described by a finite set of scenarios are novel we present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques

on blow up criteria of smooth solutions to the 3 d euler equations in a bounded domain abstract we show that a smooth solution of the 3 d euler equations in a bounded domain breaks down if and only if a certain norm of vorticity blows up at the same time here this norm is weaker than bmo norm

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

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

cauchy problem for a damped generalized imbq equation in this paper we prove that the cauchy problem for the following damped generalized imbq equation utt uxx uxxtt 2uxxt f u xx x r t 0 admits a unique global generalized solution in c3 0 wm p r l r l2 r 1 p m 0 and a unique global classical solution in c3 0 wm p r l r l2 r m 2 1p moreover the blow up of the solution for the cauchy problem of damped generalized imbq equation is studied we also prove that the cauchy problem of the above mentioned equation has a unique global generalized solution in c2 0 hs r s 12 and a unique global classical solution in c2 0 hs r s 52 and discuss the blow up of the solution

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

continuity properties of the solution map for the generalized reduced ostrovsky equation abstract it is shown that the data to solution map for the generalized reduced ostrovsky gro equation is not uniformly continuous on bounded sets in sobolev spaces on the circle with exponent s 3 2 considering that for this range of exponents the gro equation is well posed with continuous dependence on initial data this result makes the continuity of the solution map an optimal property however if a weaker h r topology is used then it is shown that the solution map becomes holder continuous in h s

vanishing viscosity for non homogeneous asymmetric fluids in r3 we consider a non homogeneous viscous incompressible asymmetric fluid in r 3 we prove that there exists a small time interval where the fluid variables converge uniformly as the viscosities tend to zero in the limit we find a non homogeneous non viscous incompressible asymmetric fluid governed by an euler like system

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

weak solutions with improved regularity for the nonhomogeneous asymmetric fluids equations with vacuum abstract we prove the existence of local and global in time weak solutions for the equations of nonhomogeneous incompressible asymmetric fluids in a bounded domain r 3 when the initial vacuum is allowed i e the initial density 0 is not necessarily strictly positive our solutions here are more regular than previous results and so satisfy the initial conditions in a stronger sense the global in time solutions are obtained under smallness assumptions on the initial data and forcing terms