strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

the initial value problem for the compressible magnetohydrodynamic equations without heat conductivity abstract the cauchy problem for the three dimensional compressible magnetohydrodynamic equations without heat conductivity is considered in this paper the global existence and uniqueness of the strong solutions are established by the delicate energy method under the condition that the initial data is small perturbations of some given constant state in h 2 framework furthermore the optimal convergence rates of the solutions in l q norm with 2 q 6 and optimal convergence rates of their spatial derivatives in l 2 norm are obtained if additionally the initial data belongs to l p with 1 p 6 5

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

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

bounds on the kth eigenvalues of trees and forests abstract we prove the upper bound k t n k 1 for the kth largest eigenvalue of a tree t 1 k n 2 and show that this bound is best possible when n 0 mod k we further obtain the strict inequality t 1 k 1 t 1 k t t 1 where t n k for the case n 0 mod k 2 k n 2 our upper bound is also proved to be the best possible upper bound for the kth largest eigenvalues of the forests in all cases

global solutions of nonlinear magnetohydrodynamics with large initial data a free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated the existence uniqueness and regularity of global solutions are established with large initial data in h1 it is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time although there is a complex interaction between the hydrodynamic and magnetodynamic effects an existence theorem of global solutions with large discontinuous initial data is also established

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

global existence and optimal decay rates for the timoshenko system the case of equal wave speeds abstract we first show the global existence and optimal decay rates of solutions to the classical timoshenko system in the framework of besov spaces due to the non symmetric dissipation the general theory for dissipative hyperbolic systems see 31 cannot be applied to the timoshenko system directly in the case of equal wave speeds we construct global solutions to the cauchy problem pertaining to data in the spatially besov spaces furthermore the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis consequently the optimal decay estimates of solution and its derivatives of fractional order are shown by time weighted energy approaches in terms of low frequency and high frequency decompositions as a by product the usual decay estimate of l 1 r l 2 r type is also shown

numerical investigation of cavitation in multidimensional compressible flows the compressible navier stokes equations for an ideal polytropic gas are considered in r n n 2 3 the question of possible vacuum formation an open theoretical problem is investigated numerically using highly accurate computational methods the flow is assumed to be symmetric about the origin with a purely radial velocity field the numerical results indicate that there are weak solutions to the navier stokes system in two and three space dimensions which display formation of vacuum when the initial data are discontinuous and sufficiently large the initial density is constant while the initial velocity field is symmetric points radially away from the origin and belongs to h s loc for all s n 2 in addition in the one dimensional case the numerical solutions are in agreement with known theoretical results

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

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

the large time behavior of solutions in the critical l p framework for compressible viscous and heat conductive gas flows in this paper we are concerned with non isentropic navier stokes equations governing compressible viscous and heat conductive gases we formulate an additional regularity assumption of low frequencies in rd d 3 and then establish the sharp time weighted inequality which allows us to get the time decay estimates of global strong solutions in the lp critical besov spaces precisely it is shown that if the initial data belong to some besov space b2 s1 with s1 1 d2 s0 s0 2dp d2 then the lp norm of the critical global solutions admits the time decay t s12 d2 12 1p in particular t d2p if s1 s0 which coincides with that of the heat kernel in the lp framework in comparison with the work of danchin and xu j math fluid mech 20 1641 1665 2018 the low frequency assumption is improved such that the regularity s1 belongs to the whole range 1 d2 s0

on the kth laplacian eigenvalues of trees with perfect matchings abstract let t n be the set of all trees of order n with perfect matchings in this paper we prove that for any tree t t n its k th largest laplacian eigenvalue k t satisfies k t 2 when n 2 k and k t n 2 k 2 n 2 k 2 4 2 when n 2 k moreover this upper bound is sharp when n 0 mod 2 k

derivative computations for the log likelihood function this note improves on a method for computing derivatives of the log likelihood function used in the identification of discrete linear time invariant gaussian systems a formula for the derivative is obtained in terms of the derivatives of the matrices used to define the system under identification

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

well posedness in critical spaces for incompressible viscoelastic fluid system abstract in this paper we study the well posedness in critical spaces of incompressible viscoelastic fluid system of oldroyd model precisely for this system with initial data in besov space of critical regularity we prove the existence and uniqueness of the local solution which is also shown to exist globally in time provided the initial data is small under certain norms

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

analytic first and second derivatives for the recursive prediction error algorithm s log likelihood function this note improves on a method for computing analytic derivatives of the likelihood function of a discrete linear time invariant gaussian state space system and extends it to handle correlation between the state transition and measurement noise terms and to compute the analytic hessian matrix

on one dimensional compressible navier stokes equations with degenerate viscosity and constant state at far fields in this paper we are concerned with the cauchy problem for one dimensional compressible isentropic navier stokes equations with density dependent viscosity 0 and pressure p 1 we will establish the global existence and asymptotic behavior of weak solutions for any 0 and 1 under the assumption that the density function keeps a constant state at far fields in particular in the case that 0 12 we obtain the large time behavior of the strong solution obtained by mellet and vasseur when the solution has a lower bound no vacuum

compressible navier stokes system large solutions and incompressible limit abstract here we prove the existence of global in time regular solutions to the two dimensional compressible navier stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density 0 for large volume bulk viscosity the result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible navier stokes equations supplemented with the divergence free projection of v 0 is global the systems are examined in r d with d 2 in the critical b 2 1 s besov spaces framework

a low frequency assumption for optimal time decay estimates to the compressible navier stokes equations the global existence issue in critical spaces for compressible navier stokes equations was addressed by danchin invent math 141 579 614 2000 and then developed by charve and danchin arch rational mech anal 198 233 271 2010 chen et al commun pure appl math 63 1173 1224 2010 and haspot arch rational mech anal 202 427 460 2011 in more general lp setting the main aim of this paper is to exhibit more precisely time decay estimates of solutions constructed in the critical regularity framework to the best of our knowledge the low frequency assumption usually plays a key role in the large time asymptotics of solutions which was firstly observed by matsumura and nishida j math kyoto univ 20 67 104 1980 in the l 1 mathbb r d space we now claim a new low frequency assumption for barotropic compressible navier stokes equations which may be of interest in the mathematical analysis of viscous fluids precisely if the initial density and velocity additionally belong to some besov space dot b sigma 1 2 infty mathbb r d with the regularity sigma 1 in 1 d 2 2d p d 2 then a sharp time weighted inequality including enough time decay information can be available where optimal decay exponents for the high frequencies are exhibited the proof mainly depends on some non standard besov product estimates as a by product those optimal time decay rates of lq lr type are also captured in the critical framework

global well posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible navier stokes equations we establish the global existence and uniqueness of classical solutions to the cauchy problem for the isentropic compressible navier stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum the initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large in particular the initial density can even have compact support these results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states 2012 wiley periodicals inc

a sharp time weighted inequality for the compressible navier stokes poisson system in the critical l p framework abstract the compressible navier stokes poisson system takes the form of usual navier stokes equations coupled with the self consistent poisson equation which is used to simulate the transport of charged particles under the electrostatic potential force in this paper we focus on the large time behavior of global strong solutions in the l p critical besov spaces inspired by the dissipative effect arising from poisson potential we formulate a new regularity assumption of low frequencies and then establish the sharp time weighted inequality which leads to the optimal time decay estimates of strong solutions indeed we see that the decay of density is faster at the half rate than that of velocity which is a different ingredient in comparison with the situation of compressible navier stokes equations our proof mainly depends on tricky and non classical besov product estimates with respect to various sobolev embeddings

global strong solution for the density dependent incompressible viscoelastic fluids in the critical lp framework abstract in this paper we consider the cauchy problem of the density dependent incompressible viscoelastic fluids in the whole space r d d 2 with the initial data close to a stable equilibrium this work is dedicated to deriving the existence result of the global strong solution in the critical space with respect to the natural scaling of the equations in the l p framework

incompressible limit of oldroyd b fluids in the whole space abstract this paper is dedicated to the study of compressible oldroyd b model in the framework of critical spaces we first prove the local well posedness for the compressible oldroyd b model with initial data 0 1 u 0 0 b 2 1 d 2 b 2 1 d 2 1 d b 2 1 d 2 d d the global well posedness is also established provided the initial data and coupling constant are sufficiently small on the basis of this we also consider the incompressible limit problem for ill prepared initial data we prove that as the mach number tends to zero the global solution to the compressible oldroyd b fluids converges to the solution to the corresponding incompressible model in some function spaces in the meanwhile the converge rates are obtained in some sense to fulfill our proof we give a new commutator estimate

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

the r bounded operator families arising from the study of the barotropic compressible flows with free surface abstract in this paper we study some model problem associated to the free boundary value problem of the barotropic compressible navier stokes equations in general smooth domain with taking surface tension into account to obtain the maximal l p l q regularity property of the model problem we prove the existence of r bounded operator families of the resolvent problem via weis theory on operator valued fourier multipliers

global well posedness of the cauchy problem of two dimensional compressible navier stokes equations in weighted spaces abstract in this paper we study the global well posedness of classical solution to 2d cauchy problem of the compressible navier stokes equations with large initial data and vacuum it is proved that if the shear viscosity is a positive constant and the bulk viscosity is the power function of the density that is with 3 then the 2d cauchy problem of the compressible navier stokes equations on the whole space r 2 admits a unique global classical solution u which may contain vacuums in an open set of r 2 note that the initial data can be arbitrarily large to contain vacuum states various weighted estimates of the density and velocity are obtained in this paper and these self contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow

global well posedness in spatially critical besov space for the boltzmann equation the unique global strong solution in the chemin lerner type space to the cauchy problem on the boltzmann equation for hard potentials is constructed in a perturbation framework such a solution space is of critical regularity with respect to the spatial variable and it can capture the intrinsic properties of the boltzmann equation for the proof of global well posedness we develop some new estimates on the nonlinear collision term through the littlewood paley theory

global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data abstract we investigate the global strong solutions for a system related to the incompressible viscoelastic fluids of oldroyd b type with a class of large initial data e 0 e 0 i j n n v 0 v 01 v 0 n we obtain the global existence and uniqueness of a solution provided that e 0 h v 0 h are sufficiently small in b 2 1 n 2 1 b 2 1 n 2 b 2 1 n 2 1 while e 0 n and v 0 n are large where e 0 h e 0 i j n 1 n v 0 h v 01 v 0 n 1 and e 0 n e 0 n 1 e 0 n n

the cauchy problem for the compressible navier stokes equations in the l p framework the global in time existence of regular solutions to the compressible navier stokes equations in the whole space is obtained the solutions are close to nontrivial equilibrium solutions moreover the result is sharp in the lp framework the velocity of the fluid belongs to wr2 1 and the initial density to w1r with r 3

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

note on the kth laplacian eigenvalues of trees with perfect matchings abstract denote by k t the k th laplacian eigenvalue of tree t let t n be the set of all trees of order n with perfect matchings in this paper we characterize the tree t in t 4 t with 2 t t 2 t 2 4 2 and the tree in t 6 t with 3 t t 2 t 2 4 2 these results affirm problem of j x li w c shiu a chang 2010 5 for the cases k 2 and k 3 respectively

global existence in critical spaces for flows of compressible viscous and heat conductive gases we are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat conductive gases the initial data are supposed to be close to a stable equilibrium with constant density and temperature using uniform estimates for the linearized system with a convection term we get global well posedness in a functional setting invariant with respect to the scaling of the associated equations in space dimension n 3 we also show a smoothing effect on the velocity and the temperature and a decay on the difference between the density and the constant reference state these results extend a previous paper devoted to the barotropic case see 5

on the existence of local classical solutions to the navier stokes equations with degenerate viscosities in this paper we consider the three dimensional cauchy problem of compressible navier stokes equations with degenerate viscosities depending on density with the initial data containing vacuum at far fields we prove the local existence of regular solutions with some compatibility conditions the regular solution is classical due to the smoothing effect of the velocity u in this paper we consider the three dimensional cauchy problem of compressible navier stokes equations with degenerate viscosities depending on density with the initial data containing vacuum at far fields we prove the local existence of regular solutions with some compatibility conditions the regular solution is classical due to the smoothing effect of the velocity u

existence and uniqueness of small energy weak solution to multi dimensional compressible navier stokes equations with large external potential force we study the 3 d compressible navier stokes equations with an external potential force and a general non decreasing pressure we prove the global in time existence of weak solutions with small energy initial data and with densities being non negative and essentially bounded a solution may have large oscillations and contain vacuum states no smallness assumption is made on the external force nor the initial perturbation in l for density initial velocity u0 is taken to be bounded in lq for some q 6 and no further regularity assumption is imposed on u0 finally we discuss the uniqueness of weak solutions

well posedness in critical spaces for the system of compressible navier stokes in larger spaces this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n geq2 we address the question of well posedness for it large data having critical besov regularity our result improves the analysis of r danchin in cite dw of chen et al in cite c2 and of the author in cite h1 h2 inasmuch as we may take initial density in b nn p 1 with 1 leq p infty our result relies on a new a priori estimate for the velocity where we introduce a new unknown called textit effective velocity to weaken one the coupling between the density and the velocity in particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density

global well posedness of compressible navier stokes equations for some classes of large initial data we prove the global well posedness of three dimensional compressible navier stokes equations for some classes of large initial data which may have large oscillation for the density and large energy for the velocity the proof uses the special structure of the system especially the effective viscous flux

global strong solutions to radial symmetric compressible navier stokes equations with free boundary in this paper we consider the two dimensional barotropic compressible navier stokes equations with stress free boundary condition imposed on the free surface as the viscosity coefficients satisfies 2 1 we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary in particular we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive and the free boundary propagates along the particle path and expand outwards at an algebraic rate

well posedness and ill posedness of a multidimensional chemotaxis system in the critical besov spaces abstract we study the cauchy problem for a multidimensional chemotaxis system in critical besov spaces b p 1 d p 2 r d b p 1 d p 1 r d d for 1 p 2 d we prove locally well posedness for large initial data and globally well posedness for small initial data of this system and more importantly we show the ill posedness in the sense that a norm inflation phenomenon occurs for p 2 d more precisely we construct a specific initial data which can be arbitrarily small in the besov spaces meanwhile the corresponding solution u can be arbitrarily large after an arbitrarily short time

lagrange structure and dynamics for solutions to the spherically symmetric compressible navier stokes equations the compressible navier stokes system cns with density dependent viscosity coefficients is considered in multi dimension the prototype of the system is the viscous saint venat model for the motion of shallow water a spherically symmetric weak solution to the free boundary value problem for cns with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path which is continuous away from the symmetry center detailed regularity and lagrangian structure of this solution have been obtained in particular it is shown that the particle path is uniquely defined starting from any non vacuum region away from the symmetry center along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved starting from any non vacuum point at a later on time a particle path is also uniquely defined backward in time which either reaches at some initial non vacuum point or stops at a small middle time and connects continuously with vacuum in addition the free boundary is shown to expand outward at an algebraic rate in time and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up this finally leads to the formation of vacuum state almost everywhere as the time goes to infinity

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

the initial value problem for the compressible navier stokes equations without heat conductivity abstract in this paper we are concerned with the global existence and convergence rates of strong solutions for the compressible navier stokes equations without heat conductivity in r 3 the global existence and uniqueness of strong solutions are established by the delicate energy method under the condition that the initial data are close to the constant equilibrium state in h 2 framework furthermore if additionally the initial data belong to l 1 the optimal convergence rates of the solutions in l 2 norm and convergence rates of their spatial derivatives in l 2 norm are obtained

global small solutions to an mhd type system the three dimensional case in this paper we consider the global well posedness of a three dimensional incompressible mhd type system with smooth initial data that is close to some nontrivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of in the coupled equations and only the horizontal derivatives of is dissipative with respect to time to achieve this we first employ anisotropic littlewood paley analysis to establish the key l1 lip 3 estimate to the third component of the velocity field then we prove the global well posedness to this system by the energy method which depends crucially on the divergence free condition of the velocity field 2014 wiley periodicals inc

on a lagrangian method for the convergence from a non local to a local korteweg capillary fluid model in the present article we are interested in further investigations for the barotropic compressible navier stokes system endowed with a non local capillarity we studied in 7 thanks to an accurate study of the associated linear system using a lagrangian change of coordinates we provide more precise energy estimates in terms of hybrid besov spaces naturally depending on a threshold frequency which is determined in function of the physical parameter distinguishing the low and the high regimes it allows us in particular to prove the convergence of the solutions from the non local to the local korteweg system another mathematical interest of this article is the study of the effect of the lagrangian change on the non local capillary term

an improved log likelihood gradient for continuous time stochastic systems with deterministic input using a covariance operator approach we derive an explicit expression for the log likelihood ratio gradient for system parameter estimation for continuous time stochastic systems with deterministic inputs the gradient formula includes the smoother estimates and derivatives of system matrices with no derivatives of estimates or covariance matrices a deterministic input is also permitted and the state noise covariance is not required to be nonsingular stable numerical techniques to calculate the gradient are also discussed

on the influence of gravity on density dependent incompressible periodic fluids abstract the present work is devoted to the analysis of density dependent incompressible fluids in a 3d torus when the froude number e goes to zero we consider the very general case where the initial data do not have a zero horizontal average where we only have smoothing effect on the velocity but not on the density and where we can have resonant phenomena on the domain we explicitly determine the limit system when e 0 and prove its global wellposedness finally we prove that for large initial data the density dependent incompressible fluid system is globally wellposed provided that e is small enough

on the role of the besov spaces for the solutions of the generalized burgers equation in homogeneous sobolev spaces abstract the generalized burgers equation tu xxu xuk 1 0 with initial data u0 in homogeneous sobolev spaces are investigated the starting point of this work is the construction of solutions in l 0 b 1 k for small size initial data in b 1 k if in addition the initial data belongs to l p s then the obtained solution is actually in l 0 l p s

zero mach number limit for compressible flows with periodic boundary conditions we address the question of convergence to the incompressible navier stokes equations for slightly compressible viscous flows with ill prepared initial data and periodic boundary conditions the case of the whole space has been studied in an earlier paper by the author the functional setting is very close to the one used by h fujita and t kato for incompressible flows for arbitrarily large initial data we show that the compressible flow with small mach number exists as long as the incompressible one does in particular it exists globally if the corresponding incompressible solution exists for all time we further state a convergence result for the slightly compressible solution filtered by the group of acoustics introduction an engineer would hardly make a distinction between incom pressible flows and slightly compressible flows indeed incompressible navier stokes equations are usually regarded as an appropriate model for describing water in the present paper we study whether such an approximation is relevant from a mathematical viewpoint in the case of periodic boundary conditions the case of the whole space has been treated in 8 after some convenient rescaling see for instance the appendix of 16 or the introduction of 21 the equations for a slightly compressible fluid read

on complementary models and fixed interval smoothing a new algorithm is derived for the standard fixed interval linear smoothing problem in which the signal is generated by a state model the structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix pi 0 since unlike in existing algorifiuns the relevant riccati equation is entirely independent of pi 0 the derivation of the algorithm is based on properties of complementary models

global variational solutions to the compressible magnetohydrodynamic equations abstract we prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic mhd equations with the coefficients depending on the temperature as a by product we give a simple proof for the nonexistence of nontrivial weak time periodic solutions by the entropy principle of clausius duhem and a new poincare type inequality

local in time results for local and non local capillary navier stokes systems with large data abstract in this article we study three capillary compressible models the classical local navier stokes korteweg system and two non local models for large initial data bounded away from zero and with a reference pressure state which is not necessarily stable p can be non positive we prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non local models towards the local korteweg model the results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients

a new method for evaluating the log likelihood gradient score of linear dynamic systems a novel method is presented that is based on the optimal smoothing equations the result can be used for efficient calculations and approximations of gradient search algorithms for maximum likelihood estimation of the unknown system parameters the method is applied to the continuous discrete case

global low energy weak solutions of the equations of three dimensional compressible magnetohydrodynamics we prove the global in time existence of weak solutions of the equations of compressible magnetohydrodynamics in three space dimensions with initial data small in l2 and initial density positive and essentially bounded a great deal of information concerning partial regularity is obtained velocity vorticity and magnetic field become relatively smooth in positive time h1 but not h2 and singularities in the pressure cancel those in a certain multiple of the divergence of the velocity thus giving concrete expression to conclusions obtained formally from the rankine hugoniot conditions

proof of a problem on laplacian eigenvalues of trees abstract denote by k l t the k th laplacian eigenvalue of a tree t let t k 2 t be the set of all trees of order 2 t k with perfect matchings in this note the trees t in t k 2 t with k l t t 2 t 2 4 2 are characterized which solves problem of j x li w c shiu and a chang in 3 completely

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

the cauchy problem for the pressureless euler isentropic navier stokes equations we present a new hydrodynamic model consisting of the pressureless euler equations and the isentropic compressible navier stokes equations where the coupling of two systems is through the drag force this coupled system can be derived in the hydrodynamic limit from the particle fluid equations that are frequently used to study the medical sprays aerosols and sedimentation problems for the proposed system we first construct the local in time classical solutions in an appropriate l2l2 sobolev space we also establish the a priori large time behavior estimate by constructing a lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities and using this together with the bootstrapping argument we obtain the global classical solution the large time behavior estimate asserts that the velocity functions of the pressureless euler and the compressible navier stokes equations are aligned exponentially fast as time tends to infinity

well posedness of a 3d parabolic hyperbolic keller segel system in the sobolev space framework abstract we investigate global strong solution to a 3 dimensional parabolic hyperbolic system arising from the keller segel model we establish the global well posedness and asymptotic behavior in the energy functional setting precisely speaking if the initial difference between cell density and its mean is small in l 2 and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in h 1 then they remain to be small in l 2 h 1 for all time moreover if the mean value of the initial cell density is smaller than some constant then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity the proof relies on an application of fourier analysis to a linearized parabolic hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration

global well posedness for compressible viscoelastic fluids near equilibrium in this paper we prove local well posedness for compressible viscoelastic fluids of the oldroyd model under the assumption that the initial density is bounded away from zero and global well posedness near equilibrium the proof of global well posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic parabolic system with convection terms

global existence of strong solution for viscous shallow water system with large initial data on the irrotational part abstract we are interested in studying the cauchy problem for the viscous shallow water system in dimension n 2 we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations more precisely our smallness assumption on the initial data is supercritical for the scaling of the equations it allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension n 2 to do this we introduce the notion of quasi solutions which consists in solving the pressureless viscous shallow water system we can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density this smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the quasi solutions indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations to finish we prove the existence of global strong solution with large initial data when n 2 provided that the mach number is sufficiently large

global existence and optimal decay rate for the strong solutions in h2 to the 3 d compressible navier stokes equations without heat conductivity abstract in this paper we consider the 3 d compressible navier stokes equations without heat conductivity which form a hyperbolic parabolic system we prove the global existence of a strong solution when the initial perturbation is small in h 2 and its l 1 norm is bounded moreover we can obtain the optimal decay rates for such a solution

a new formula for the log likelihood gradient for continuous time stochastic systems using a finitely additive white noise approach we obtain an explicit expression for the gradient of the log likelihood ratio for system parameter estimation for continuous time linear stochastic systems with noisy observations our gradient formula includes the smoother estimates of the state vector and derivatives of only the system matrices and not the estimates or error covariances a scheme to calculate the log likelihood gradient without solving a riccati equation is described when only one matrix and the initial covariance depend on the unknown parameter

global existence and minimal decay regularity for the timoshenko system the case of non equal wave speeds abstract as a continued work of 18 we are concerned with the timoshenko system in the case of non equal wave speeds which admits the dissipative structure of regularity loss firstly with the modification of a priori estimates in 18 we construct global solutions to the timoshenko system pertaining to data in the besov space with the regularity s 3 2 owing to the weaker dissipative mechanism extra higher regularity than that for the global in time existence is usually imposed to obtain the optimal decay rates of classical solutions so it is almost impossible to obtain the optimal decay rates in the critical space to overcome the outstanding difficulty we develop a new frequency localization time decay inequality which captures the information related to the integrability at the high frequency part furthermore by the energy approach in terms of high frequency and low frequency decomposition we show the optimal decay rate for timoshenko system in critical besov spaces which improves previous works greatly

global existence for the multi dimensional compressible viscoelastic flows abstract the global solutions in critical spaces to the multi dimensional compressible viscoelastic flows are considered the global existence of the cauchy problem with initial data close to an equilibrium state is established in besov spaces using uniform estimates for a hyperbolic parabolic linear system with convection terms we prove the global existence in the besov space which is invariant with respect to the scaling of the associated equations several important estimates are achieved including a smoothing effect on the velocity and the l 1 decay of the density and deformation gradient

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

some uniform estimates and blowup behavior of global strong solutions to the stokes approximation equations for two dimensional compressible flows abstract this paper concerns the global existence and the large time behavior of strong and classical solutions to the two dimensional 2d stokes approximation equations for the compressible flows we consider the unique global strong solution or classical solution to the 2d stokes approximation equations for the compressible flows together with the space periodicity boundary condition or the no stick boundary condition or cauchy problem for arbitrarily large initial data first we prove that the density is bounded from above independent of time in all these cases secondly we show that for the space periodicity boundary condition or the no stick boundary condition if the initial density contains vacuum at least at one point then the global strong or classical solution must blow up as time goes to infinity

optimal decay rate for strong solutions in critical spaces to the compressible navier stokes equations abstract in this paper we are concerned with the convergence rates of the global strong solution to motionless state with constant density for the compressible navier stokes equations in the whole space r n for n 3 it is proved that the perturbations decay in critical spaces if the initial perturbations of density and velocity are small in b 2 1 n 2 r n b 1 0 r n and b 2 1 n 2 1 r n b 1 0 r n respectively

global well posedness for compressible navier stokes equations with highly oscillating initial velocity in this paper we prove global well posedness for compressible navier stokes equations in the critical functional framework with the initial data close to a stable equilibrium this result allows us to construct global solutions for the highly oscillating initial velocity the proof relies on a new estimate for the hyperbolic parabolic system with convection terms 2010 wiley periodicals inc

compressible navier stokes equations with zero heat conductivity abstract we study the large time behavior of solutions of the compressible navier stokes equations without heat conductivity acoustic waves are dissipative due to the viscosity entropy waves are nondissipative due to lack of heat conductivity thus the system is of composite type the simplest prototype of this class of nonlinear partial differential equations we study the nonlinear interactions of waves using a combination of energy method weighted energy method and pointwise estimates

gradient of the log likelihood ratio for infinite dimensional stochastic systems using a covariance operator approach we derive an expression for the log likelihood ratio gradient for system parameter estimation for continuous time infinite dimensional stochastic systems the gradient formula includes the smoother estimates and derivatives of system operators with no derivatives of estimates or covariance operators the unbounded operators typically found in partial differential equations limit how much the gradient formula can be simplified a random heat equation is considered

analyticity of solutions to the barotropic compressible navier stokes equations abstract in this paper we establish analyticity of solutions to the barotropic compressible navier stokes equations describing the motion of the density and the velocity field u in r 3 we assume that 0 is a small perturbation of 1 and 1 1 0 u 0 are analytic in besov spaces with analyticity radius 0 we show that the corresponding solutions are analytic globally in time when 1 1 0 u 0 are sufficiently small to do this we introduce the exponential operator e t d acting on 1 1 u where d is the differential operator whose fourier symbol is given by 1 1 2 3 and t is chosen to satisfy t globally in time

the incompressible limit in l p type critical spaces this paper aims at justifying the low mach number convergence to the incompressible navier stokes equations for viscous compressible flows in the ill prepared data case the fluid domain is either the whole space or the torus a number of works have been dedicated to this classical issue all of them being to our knowledge related to l 2 spaces and to energy type arguments in the present paper we investigate the low mach number convergence in the l p type critical regularity framework more precisely in the barotropic case the divergence free part of the initial velocity field just has to be bounded in the critical besov space dot b d p 1 p r cap dot b 1 infty 1 for some suitable p r in 2 4 times 1 infty we still require l 2 type bounds on the low frequencies of the potential part of the velocity and on the density though an assumption which seems to be unavoidable in the ill prepared data framework because of acoustic waves in the last part of the paper our results are extended to the full navier stokes system for heat conducting fluids

non existence of classical solutions with finite energy to the cauchy problem of the compressible navier stokes equations the well posedness of classical solutions with finite energy to the compressible navier stokes equations cns subject to arbitrarily large and smooth initial data is a challenging problem in the case when the fluid density is away from vacuum strictly positive this problem was first solved for the cns in either one dimension for general smooth initial data or multi dimension for smooth initial data near some equilibrium state that is small perturbation antontsev et al in boundary value problems in mechanics of nonhomogeneous fluids north holland publishing co amsterdam 1990 kazhikhov in sibirsk mat zh 23 60 64 1982 kazhikhov et al in prikl mat meh 41 282 291 1977 matsumura and nishida in proc jpn acad ser a math sci 55 337 342 1979 j math kyoto univ 20 67 104 1980 commun math phys 89 445 464 1983 in the case that the flow density may contain a vacuum the density can be zero at some space time point it seems to be a rather subtle problem to deal with the well posedness problem for cns the local well posedness of classical solutions containing a vacuum was shown in homogeneous sobolev space without the information of velocity in l2 norm for general regular initial data with some compatibility conditions being satisfied initially cho et al in j math pures appl 9 83 243 275 2004 cho and kim in j differ equ 228 377 411 2006 manuscr math 120 91 129 2006 choe and kim in j differ equ 190 504 523 2003 and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations huang et al in commun pure appl math 65 549 585 2012 however it was shown that any classical solutions to the compressible navier stokes equations in finite energy inhomogeneous sobolev space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported xin in commun pure appl math 51 229 240 1998 in this paper we investigate the well posedess of classical solutions to the cauchy problem of navier stokes equations and prove that the classical solution with finite energy does not exist in the inhomogeneous sobolev space for any short time under some natural assumptions on initial data near the vacuum this implies in particular that the homogeneous sobolev space is as crucial as studying the well posedness for the cauchy problem of compressible navier stokes equations in the presence of a vacuum at far fields even locally in time

global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations abstract in this paper we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in r 3 we prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in h 3 framework moreover if additionally the initial data belong to l p with 1 p 6 5 the optimal convergence rates of the solutions in l q norm with 2 q 6 and its spatial derivatives in l 2 norm are obtained