brezis gallouet wainger type inequalities and blow up criteria for navier stokes equations in unbounded domains we shall find the weakest norm that satisfies the brezis gallouet wainger type inequality under some conditions as an application of the brezis gallouet wainger type inequality we shall establish beale kato majda type blow up criteria of smooth solutions to the 3 d navier stokes equations in unbounded domains

global well posedness of incompressible inhomogeneous fluid systems with bounded density or non lipschitz velocity in this paper we first prove the global existence of weak solutions to the d dimensional incompressible inhomogeneous navier stokes equations with initial data a 0 in l infty mathbb r d u 0 u 0 h u 0 d in dot b 1 frac d p p r mathbb r d which satisfy mu a 0 l infty u 0 h dot b 1 frac d p p r rm exp c r mu 2r u 0 d dot b 1 frac d p p r 2r leqq c 0 mu for some positive constants c 0 c r and 1 p d 1 r the regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non lipschitz velocity fields furthermore with additional regularity assumptions on the initial velocity or on the initial density we can also prove the uniqueness of such a solution we should mention that the classical maximal l p l q regularity theorem for the heat kernel plays an essential role in this context

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

an optimization algorithm for the vehicle routing problem with time windows based on lagrangian relaxation our paper presents a new optimization method for the vehicle routing problem with time windows vrptw the vrptw is a generalization of the vehicle routing problem where the service of a customer must start within a given time interval a so called time window our method is based on a lagrangian relaxation of the constraint set requiring that each customer must be serviced the master problem consists of finding the optimal lagrangian multipliers and the subproblem is a shortest path problem with time windows and capacity constraints the optimal multipliers are found using a method exploiting the benefits of subgradient methods as well as a bundle method the method has been implemented and tested on a series of well known benchmark problems of size up to 100 customers our algorithm turns out to be very competitive compared to algorithms considered in the literature and we have succeeded in solving several previously unsolved problems

local existence and uniqueness of strong solutions to the navier stokes equations with nonnegative density abstract in this paper we consider the initial boundary value problem to the nonhomogeneous incompressible navier stokes equations local strong solutions are established for any initial data 0 u 0 w 1 l h 0 1 with 1 and if 2 then the strong solution is unique the initial density is allowed to be nonnegative and in particular the initial vacuum is allowed the assumption on the initial data is weaker than the previous widely used one that 0 u 0 h 1 l h 0 1 h 2 and no compatibility condition is required

a lagrangian approach for the incompressible navier stokes equations with variable density we investigate the cauchy problem for the inhomogeneous navier stokes equations in the whole n dimensional space under some smallness assumption on the data we show the existence of global in time unique solutions in a critical functional framework the initial density is required to belong to the multiplier space of input amssym dot b n p 1 p 1 bbb r n in particular piecewise constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities using lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem as a consequence conditions for uniqueness are the same as for existence 2012 wiley periodicals inc

incompressible flows with piecewise constant density we investigate the incompressible navier stokes equations with variable density the aim is to prove existence and uniqueness results in the case of discontinuous initial density in dimension n 2 3 assuming only that the initial density is bounded and bounded away from zero and that the initial velocity is smooth enough we get the local in time existence of unique solutions uniqueness holds in any dimension and for a wider class of velocity fields in particular all those results are true for piecewise constant densities with arbitrarily large jumps global results are established in dimension two if the density is close enough to a positive constant and in n dimensions if in addition the initial velocity is small the lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper

regularity criteria for the 3d magneto micropolar fluid equations in besov spaces with negative indices abstract we consider the cauchy problem of the magneto micropolar fluid equations in three space dimensions it is proved that if the velocity magnetic field and the micro rotational velocity belong to some critical besov space with negative indices then the solution is in fact smooth

on the well posedness of the full compressible navier stokes system in critical besov spaces abstract we are concerned with the cauchy problem of the full compressible navier stokes equations satisfied by viscous and heat conducting fluids in r n we focus on the so called critical besov regularity framework in this setting it is natural to consider initial densities 0 velocity fields u 0 and temperatures 0 with a 0 0 1 b p 1 n p u 0 b p 1 n p 1 and 0 b p 1 n p 2 after recasting the whole system in lagrangian coordinates and working with the total energy along the flow rather than with the temperature we discover that the system may be solved by means of banach fixed point theorem in a critical functional framework whenever the space dimension is n 2 and 1 p 2 n back to eulerian coordinates this allows to improve the range of p s for which the system is locally well posed compared to 7

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

a heuristic method for the set covering problem we present a lagrangian based heuristic for the well known set covering problem scp the algorithm was initially designed for solving very large scale scp instances involving up to 5 000 rows an

exact results and self averaging properties for random random walks on a one dimensional infinite lattice we present new exact results for a one dimensional asymmetric disordered hopping model the lattice is taken infinite from the start and we do not resort to the periodization scheme used by derrida an explicit resummation allows for the calculation of the velocityv and the diffusion constantd which are found to coincide with those given by derrida and for demonstrating thatv is indeed a self averaging quantity the same property is established ford in the limiting case of a directed walk

boussinesq system with measure forcing the paper analyzes the navier stokes system coupled with the convective diffusion equation responsible for thermal effects it is a version of the boussinesq approximation with a heat source the problem is studied in the two dimensional plane and the heat source is a measure transported by the flow for arbitrarily large initial data we prove global in time existence of unique regular solutions measure being a heat source limits regularity of constructed solutions and it requires a non standard framework of inhomogeneous besov spaces of the l infty 0 t b s p infty type the uniqueness of solutions is proved by using the lagrangian coordinates

existence theorem and blow up criterion of the strong solutions to the two fluid mhd equation in r3 abstract we first give the local well posedness of strong solutions to the cauchy problem of the 3d two fluid mhd equations and then study the blow up criterion of the strong solutions by means of the fourier frequency localization and bony s paraproduct decomposition it is proved that the strong solution u b can be extended after t t if either u l t q b p 0 with 2 q 3 p 1 and b l t 1 b 0 or j l t q b p 0 with 2 q 3 p 2 where t u denotes the vorticity of the velocity and j b stands for the current density

global optimality conditions for discrete and nonconvex optimizationwith applications to lagrangian heuristics and column generation the well known and established global optimality conditions based on the lagrangian formulation of an optimization problem are consistent if and only if the duality gap is zero we develop a set of global optimality conditions that are structurally similar but are consistent for any size of the duality gap this system characterizes a primaldual optimal solution by means of primal and dual feasibility primal lagrangian e optimality and in the presence of inequality constraints a relaxed complementarity condition analogously called complementarity the total size e of those two perturbations equals the size of the duality gap at an optimal solution further the characterization is equivalent to a near saddle point condition which generalizes the classic saddle point characterization of a primaldual optimal solution in convex programming the system developed can be used to explain to a large degree when and why lagrangian heuristics for discrete optimization are successful in reaching near optimal solutions further experiments on a set covering problem illustrate how the new optimality conditions can be utilized as a foundation for the construction of new lagrangian heuristics finally we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

global well posedness to some modified 2 d nonhomogeneous navier stokes equations in this paper we are concerned with the global well posedness of modified 2d nonhomogeneous navier stokes equations our results improve the results of zhang int math res not 2008 1 rnn098 to some extent in this paper we are concerned with the global well posedness of modified 2d nonhomogeneous navier stokes equations our results improve the results of zhang int math res not 2008 1 rnn098 to some extent

on the global well posedness of 2 d inhomogeneous incompressible navier stokes system with variable viscous coefficient abstract given solenoidal vector u 0 h 2 h 1 r 2 0 1 l 2 r 2 and 0 l w 1 r r 2 with a positive lower bound for 0 1 2 and 2 r 2 1 2 we prove that 2 d incompressible inhomogeneous navier stokes system 1 1 has a unique global solution provided that the viscous coefficient 0 is close enough to 1 in the l norm compared to the size of and the norms of the initial data with smoother initial data we can prove the propagation of regularities for such solutions furthermore for 1 p 4 if 0 1 u 0 belongs to the critical besov spaces b p 1 2 p r 2 b p 1 1 2 p l 2 r 2 and the b p 1 2 p r 2 norm of 0 1 is sufficiently small compared to the exponential of u 0 l 2 2 u 0 b p 1 1 2 p we prove the global well posedness of 1 1 in the scaling invariant spaces finally for initial data in the almost critical besov spaces we prove the global well posedness of 1 1 under the assumption that the l norm of 0 1 is sufficiently small

intermittency in random maps abstract the escape probability for a random walk on a one dimensional lattice is discussed in terms of random maps the global dynamics is found to be intermittent the laminar regions with long sojourn near the upper fixed point are broken by irregular bursts intermittency emerges even if the sinai condition is not satisfied it is shown that if p is the probability of choosing the upper map the average laminar length diverges as 1 p

analytic study of a model of diffusion on a random comblike structure abstract we present an analytic study of a model of diffusion on a random comblike structure in which no bias field does exist neither along the backbone nor along the branches for any given disordered structure our analytic treatment allows to compute in an exact manner the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone and of the particle position and dispersion the expressions of these quantities are shown to coincide asymptotically with those derived in simple mean field treatments the results for any given sample do not depend on the particular configuration self averaging property the behaviours strongly depend on the distribution of the lengths of the branches with an exponential distribution transport is normal while anomalous diffusion may take place for a power law distribution when long branches are present with a sufficiently high weight

on the well posedness of the ideal mhd equations in the triebel lizorkin spaces in this paper we prove the local well posedness for the ideal mhd equations in the triebel lizorkin spaces and obtain a blow up criterion of smooth solutions specifically we fill a gap in a step of the proof of the local well posedness part for the incompressible euler equation in chae comm pure appl math 55 654 678 2002

on the well posedness of 3 d inhomogeneous incompressible navier stokes equations with variable viscosity abstract in this paper we mainly study the well posedness for the 3 d inhomogeneous incompressible navier stokes equations with variable viscosity with some smallness assumption on the bmo norm of the initial density we first get the local well posedness of 1 1 in the critical besov spaces moreover if the viscosity coefficient is a constant we can extend this local solution to be a global one our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by abidi gui and zhang in 3 burtea in 8 and zhai and yin in 32 to approach the ideal one i e 1 p 6 the main novelty of this work is to apply the crw theorem obtained by coifman rochberg weiss in 11 to get a new a priori estimate for an elliptic equation with variable coefficients the uniqueness of the solution also relies on a lagrangian approach as in 16 17 18

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

on the global well posedness of 2 d boussinesq system with variable viscosity abstract in this paper we investigate the global well posedness of 2 d boussinesq system which has variable kinematic viscosity and with thermal conductivity of d with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in l norm

fujita kato theorem for the 3 d inhomogeneous navier stokes equations abstract in this paper we prove the global existence and uniqueness of the solution with discontinuous density for the 3 d inhomogeneous navier stokes equations if the initial data 0 u 0 l r 3 h s r 3 with s 1 2 satisfies 0 c 0 0 c 0 u 0 h 1 2 e for some small e 0 depending only on c 0 c 0 furthermore we introduce the dual method to show that if u 0 l p r 3 for p 6 5 2 the velocity satisfies the decay estimate k u t l 2 c 1 t k 2 p for t 1 k 0 1 with p 3 2 1 p 1 2

global solution to the 3d inhomogeneous nematic liquid crystal flows with variable density abstract in this paper we investigate the global existence and uniqueness of solution to the 3d inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of besov spaces it is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data 0 1 u 0 n 0 e 3 m b p 1 3 p 1 r 3 b p 1 3 p 1 r 3 b p 1 3 p r 3 with 1 p 6 and satisfies 0 1 m b p 1 3 p 1 u 0 b p 1 3 p 1 n 0 e 3 b p 1 3 p c for some small c 0 depending only on p here m b p 1 3 p 1 r 3 is the multiplier space of besov space b p 1 3 p r 3 using the lagrangian approach in danchin and mucha 2012 2013 7 8 is the key to our results

note on the complexity of the shortest path models for column generation in vrptw in this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the vrptw is justified on complexity grounds that is the first dynamic programming model presented in m desrochers j desrosiers and m solomon 1992 that is able to price out all feasible routes is np hard in the strong sense

between homogeneous and inhomogeneous navier stokes systems the issue of stability abstract we construct large velocity vector solutions to the three dimensional inhomogeneous navier stokes system the result is proved via the stability of two dimensional solutions with constant density under the assumption that initial density is point wisely close to a constant key elements of our approach are estimates in the maximal regularity regime and the lagrangian coordinates considerations are done in the whole r 3

real time vehicle rerouting problems with time windows this paper introduces and studies real time vehicle rerouting problems with time windows applicable to delivery and or pickup services that undergo service disruptions due to vehicle breakdowns in such problems one or more vehicles need to be rerouted in real time to perform uninitiated services with the objective to minimize a weighted sum of operating service cancellation and route disruption costs a lagrangian relaxation based heuristic is developed which includes an insertion based algorithm to obtain a feasible solution for the primal problem a dynamic programming based algorithm solves heuristically the shortest path problems with resource constraints that result from the lagrangian relaxation computational experiments show that the developed lagrangian heuristic performs very well

on regularity criterion for the dissipative quasi geostrophic equations abstract in this paper we consider the 2d dissipative quasi geostrophic equations and study the regularity criterion of the solutions by means of a commutator estimate based on frequency localization and bony s paraproduct decomposition we obtain a regularity criterion l r 0 t b p 0 for 2 p r 2 p which is an obvious extension of the result in b dong z chen a remark on regularity criterion for the dissipative quasi geostrophic equations j math anal appl 329 2007 1212 1217 where their restrictive condition on p is 4 p

dynamical properties of a random walk in a one dimensional random medium abstract the random motion of a particle in a one dimensional continuous random medium with random forces is investigated by making use of the large scale equivalence of the properties of the walk with those of a directed random walk on a discrete lattice when the disorder is weak enough a normal drift diffusion regime takes place whereas for strong disorder anomalous behaviours occur it is shown that these results may be obtained by introducing a renormalized lattice with a finite spacing in the normal phase this spacing appears explicitly in the result while it turns out to be irrelevant in the anomalous ones the dynamical exponents as well as the prefactors of the power laws of the particle position and of its average dispresion are explicitly calculated in the anomalous phases

a critical functional framework for the inhomogeneous navier stokes equations in the half space abstract this paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous navier stokes equations in the half space in the case of small data with critical regularity in dimension n 3 we state that if the initial density 0 is close to a positive constant in l w n 1 r n and the initial velocity u 0 is small with respect to the viscosity in the homogeneous besov space b n 1 0 r n then the equations have a unique global solution the proof strongly relies on new maximal regularity estimates for the stokes system in the half space in l 1 0 t b p 1 0 r n interesting for their own sake

incompressible inhomogeneous fluids in bounded domains of r3 with bounded density abstract in this paper we study the incompressible inhomogeneous navier stokes equations in bounded domains of r 3 involving bounded density functions 1 a based on the corresponding theory of besov spaces on domains we first obtain the global existence of weak solutions u with initial data a 0 l u 0 b q s 1 3 q for 1 q 3 1 s furthermore with additional regularity assumptions on the initial velocity we also prove the uniqueness of such a solution it is a generalization of a result established by huang et al 2013 20 for the whole space r 3

on the wellposedness of three dimensional inhomogeneous navier stokes equations in the critical spaces we prove the local wellposedness of three dimensional incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces without assumptions of small density variation furthermore if the initial velocity field is small enough in the critical besov space dot b 1 2 2 1 mathbb r 3 this system has a unique global solution

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

directed random walks in random environments we use holding time methods to study the asymptotic behavior of pure birth processes with random transition rates both the normal and slow approaches to infinity are studied fluctuations are shown to obey the central limit theorem for almost all sample transition rates our results are stronger and our proofs simpler then those of recently published studies

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

diffusion on random comblike structures field induced trapping effects an analytic study of diffusion in random comblike structures in which field induced trapping effects enter into play is presented in order to mimic at best a percolation cluster above pc the distribution of the lengths of the branches is taken as exponential the branches act as traps of efficiency growing with the applied bias the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone and of its position and dispersion are computed in an exact manner as a result when the field exceeds a certain critical value anomalous behaviours emerge characterized by exponents depending continuously on the field

a regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces abstract we consider the regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces it is proved that if u l p 0 t f q 2 q 3 0 with 2 p 3 q 2 3 2 q then the solution remains smooth in 0 t as a corollary we obtain the classical beal kato majda criterion that is the condition u l 1 0 t b 0 ensures the smoothness of the solution

global fujita kato solution of 3 d inhomogeneous incompressible navier stokes system abstract in this paper we shall prove the global existence of weak solutions to 3d inhomogeneous incompressible navier stokes system ins with initial density in the bounded function space and having a positive lower bound and with initial velocity being sufficiently small in the critical besov space b 2 1 1 2 this result corresponds to the fujita kato solutions of the classical navier stokes system the same idea can be used to prove the global existence of weak solutions in the critical functional framework to ins with one component of the initial velocity being large and can also be applied to provide a lower bound for the lifespan of smooth enough solutions of ins

logarithmically improved bkm s criterion for the 3d nematic liquid crystal flows abstract this paper proves a logarithmically improved beale kato majda s criterion for the 3d nematic liquid crystal flows which says that if 0 t u t b m o ln e u t b m o d t then the solution u d is smooth up to the time t t

remarks on the blowup criteria for oldroyd models abstract we provide a new method to prove and improve the chemin masmoudi criterion for viscoelastic systems of oldroyd type in j y chemin n masmoudi about lifespan of regular solutions of equations related to viscoelastic fluids siam j math anal 33 1 2001 84 112 in two space dimensions our method is much easier than the one based on the well known losing a priori estimate and is expected to be easily adopted to other problems involving the losing a priori estimate

global well posedness for the 3d incompressible inhomogeneous navier stokes equations and mhd equations abstract the present paper is dedicated to the global well posedness for the 3d inhomogeneous incompressible navier stokes equations in critical besov spaces without smallness assumption on the variation of the density we aim at extending the work by abidi gui and zhang 2012 2 and 2013 3 to a lower regularity index about the initial velocity the key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in besov spaces which have the same degree as l 2 in r 3 finally we also generalize our well posedness result to the inhomogeneous incompressible mhd equations

on the well posedness of 2 d incompressible navier stokes equations with variable viscosity in critical spaces abstract in this paper we first prove the local well posedness of the 2 d incompressible navier stokes equations with variable viscosity in critical besov spaces with negative regularity indices without smallness assumption on the variation of the density the key is to prove for p 1 4 and a b p 1 2 p r 2 that the solution mapping h a f to the 2 d elliptic equation div 1 a div f is bounded on b p 1 2 p 1 r 2 more precisely we prove that b p 1 2 p 1 c 1 a b p 1 2 p 2 f b p 1 2 p 1 the proof of the uniqueness of solution to 1 2 relies on a lagrangian approach 15 16 17 when the viscosity coefficient is a positive constant we prove that 1 2 is globally well posed

microscopic dynamical exponents for random random directed walk on a one dimensional lattice with quenched disorder we demonstrate that the dynamical exponent for the time dependence of the coordinate previously found for an average over disorder is already present in any realization of a given sample this ergodicity comes from the existence of a scaling law for the probability distribution of the parameter defining the asymptotic dynamical regime the self averaging or non self averaging properties of the normal or anomalous phases are direct consequences of this result

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

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

dynamical exponents for one dimensional random random directed walks the dynamical exponents of the coordinate and of the mean square displacement are explicitly calculated in the case of a directed random walk on a one dimensional random lattice moreover it is shown that in the dynamical phase where the coordinate increases slower thant the latter is not a self averaging quantity

a tabu search heuristic for the vehicle routing problem with stochastic demands and customers this paper considers a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands a tabu search heuristic is developed for this problem comparisons with known optimal solutions on problems whose sizes vary from 6 to 46 customers indicate that the heuristic produces an optimal solution in 89 45 of cases with an average deviation of 0 38 from optimality

global well posedness to the 3 d incompressible inhomogeneous navier stokes equations with a class of large velocity in this article we consider the global well posedness to the 3 d incompressible inhomogeneous navier stokes equations with a class of large velocity more precisely assuming a0 bq 13q r3 and u0 u0h u03 bp 1 1 3p r3 for p q 1 6 with sup 1p 1q 13 inf 1p 1q we prove that if ca0bq 13q u03bp 1 1 3p 1 1 c u0hbp 1 1 3p u03bp 1 1 3p1 u0hbp 1 1 3p 1 then the system has a unique global solution a c 0 bq 13q r3 u c 0 bp 1 1 3p r3 l1 r bp 11 3p r3 it improves the recent result of m paicu and p zhang j funct anal 262 3556 3584 2012 where the exponent form of the initial smallness condition is replaced by a polynomial form

global solution to the incompressible oldroyd b model in the critical l p framework the case of the non small coupling parameter this paper is dedicated to the global well posedness issue of the incompressible oldroyd b model in the whole space mathbb r d with d geqq 2 it is shown that this set of equations admits a unique global solution in a certain critical lp type besov space provided that the initial data but not necessarily the coupling parameter is small enough as a consequence even through the coupling effect between the equations of velocity u and the symmetric tensor of constrains is not small one may construct the unique global solution to the oldroyd b model for a class of large highly oscillating initial velocity the proof relies on the estimates of the linearized systems of u and u mathbb p rm div tau which may be of interest for future works this result extends the work by chemin and masmoudi siam j math anal 33 84 112 2001 to the non small coupling parameter case

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

random walk in a one dimensional random medium abstract the random walk of a particle in a one dimensional random medium is examined by means of the equivalent transfer rates technique in the discrete as well as in the continuous version of the model the probability distributions of the energy dependent equivalent transfer rates are found analytically either by a matching procedure in the discrete case or exactly in the continuous model both discrete and continuous models are shown to belong to the same universality class the average probability of presence of the particle at its initial point is then computed for a non zero global bias it decreases at large times according to a negative power law with an exponent depending on disorder and bias when there is no global bias sinai s model the decay at large times follows a logarithmic law

the incompressible navier stokes equations in vacuum we are concerned with the existence and uniqueness issue for the inhomogeneous incompressible navier stokes equations supplemented with h 1 initial velocity and only bounded nonnegative density in contrast with all the previous works on that topics we do not require regularity or positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions those solutions are global in the two dimensional case for general data and in the three dimensional case if the velocity satisfies a suitable scaling invariant smallness condition as a straightforward application we provide a complete answer to lions question in 25 page 34 concerning the evolution of a drop of incompressible viscous fluid in the vacuum