a blow up criterion for two dimensional compressible viscous heat conductive flows abstract we establish a blow up criterion in terms of the upper bound of the density and temperature for the strong solution to 2d compressible viscous heat conductive flows the initial vacuum is allowed

pointwise functional calculi supposeais a possibly unbounded closed linear operator on a banach spacex x x and f is a banach algebra of functions we introduce apointwiseffunctional calculus for a at x this is a bounded linear map from f intox with the properties that one would expect from a mapf f a x ifahad a f functional calculus howeveramay not have such a functional calculus we show that the existence of a pointwise f functional calculus is equivalent to the existence of a continuously embedded banach subspace on whichahas a global f functional calculus we characterize being pointwise generalized scalar atxand give simple sufficient conditions we also discuss the relationship between pointwise functional calculi and the many physical problems that may be modelled as an abstract cauchy problem

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

global strong solution for the korteweg system in dimension n 2 this work is devoted to prove the existence of global strong solution in dimension n 2 for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see 16 which can be used as a phase transition model in a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span t of existence of the solution this one depends on the norm of the initial data 0 v 0 the second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient 2 and on the viscosity tensor which corresponds to the shallow water case 2 div du to do this we derivate different energy estimate on the density and the effective velocity v which ensures that the strong solution can be extended beyond t the main difficulty consists in controlling the vacuum or in other words to estimate the l norm of 1 the proof relies mostly on a method introduced by de giorgi 14 see also ladyzhenskaya et al in 30 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

h functional calculus and models of nagy foias type for sectorial operators we prove that a sectorial operator admits an h functional calculus if and only if it has a functional model of nagy foias type furthermore we give a concrete formula for the characteristic function in a generalized sense of such an operator more generally this approach applies to any sectorial operator by passing to a different norm the mcintosh square function norm we also show that this quadratic norm is close to the original one in the sense that there is only a logarithmic gap between them

strong solution for korteweg system in bmo 1 rn with initial density in l in this paper we investigate the question of the existence of strong solution in finite time for the korteweg system for small initial data provided that the initial momentum 0 u 0 belongs to bmo 1 t r n for t 0 and the initial density 0 is in l r n with n 1 and far away from the vacuum this result extends the so called koch tataru theorem for the korteweg system it is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes lipschitz for any t with t 0 we also prove the existence of global strong solution for initial data 0 1 0 u 0 b n 2 1 2 r n b n 2 2 r n l r n b n 2 1 2 r n n this result allows in particular to extend the notion of oseen solution corresponding to particular solution of the incompressible navier stokes system in dimension n 2 to the korteweg system provided that the vorticity of the momentum 0 u 0 is a dirac mass 0 with sufficiently small ihowever unlike the navier stokes equations the property of self similarity is not conserved for the korteweg system since there is no invariance by scaling because the term of pressure

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

a priori estimates for the 3d quasi geostrophic system abstract the present article is devoted to the 3d dissipative quasi geostrophic system qg this system can be obtained as a limit model of the primitive equations in the asymptotics of strong rotation and stratification and involves a non radial non local homogeneous pseudo differential operator of order 2 denoted by and whose semigroup kernel reaches negative values after a refined study of the non local part of we prove a priori estimates in the general l p setting for the 3d qg model the main difficulty of this article is to study this highly non classical operator and the commutator of with a lagrangian change of variable an important application of these a priori estimates providing bound from below to the lifespan of the solutions of the primitive equations for ill prepared blowing up initial data can be found in a companion paper

discontinuous solutions of the navier stokes equations for multidimensional flows of heat conducting fluids we prove the global existence of weak solutions of the navier stokes equations for compressible heat conducting fluids in two and three space dimensions when the initial density is close to a constant in l 2 l the initial temperature is close to a constant in l 2 and the initial velocity is small in h s l 4 where s 0 when n 2 and when n 3 the l p norms must be weighted slightly when n 2 in particular the initial data may be discontinuous across a hypersurface of n a great deal of qualitative information about the solution is obtained for example we show that the velocity vorticity and temperature 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 central role in the entire analysis particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer moreover f is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself an effect which is crucial for obtaining time independent pointwise bounds for the density

computation of singularities in large nonlinear systems a general approach is proposed for the numerical computation of one parameter singularities in the sense of golubitsky and schaeffer singularities and groups in bifurcation theory vol i springer verlag berlin 1985 the singularities of codimension three or less are organized in a new way our scheme is close to the numerics and in some cases gives explicit expressions for the defining equations obtained inductively where no such expressions were previously available the numerical computation by the lyapunov schmidt reduction summarizes work by many authors during the last 20 years treating all cases in a unified way a numerical example is provided where singularities with codimension up to three are detected and computed in a parameterized boundary value problem

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

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

optimal order error estimates for finite element approximations of a bifurcation function the bifurcation function associated with an elliptic boundary value problem au g u 0 is a vector field b on rd with the property that the solutions of the boundary value problem are in a one to one correspondence with the zeros of b a finite element approximation bh of b is formulated and optimal order error estimates are derived implementation issues are also discussed

on the maximal lp lq regularity for a compressible fluid model of korteweg type on general domains abstract the aim of this paper is to show the maximal l p l q regularity for a compressible fluid model of korteweg type on general domains of the n dimensional euclidean space for n 2 e g bounded domains exterior domains half spaces layers tubes and their perturbed domains our approach is based on the theory of the r boundedness for a generalized resolvent problem associated with the korteweg type model

vanishing viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow we establish the vanishing viscosity limit of the navier stokes equations to the isentropic euler equations for one dimensional compressible fluid flow for the navier stokes equations there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup norm of solutions with respect to the physical viscosity coefficient may not be directly controllable furthermore convex entropy entropy flux pairs may not produce signed entropy dissipation measures to overcome these difficulties we first develop uniform energy type estimates with respect to the viscosity coefficient for solutions of the navier stokes equations and establish the existence of measure valued solutions of the isentropic euler equations generated by the navier stokes equations based on the uniform energy type estimates and the features of the isentropic euler equations we establish that the entropy dissipation measures of the solutions of the navier stokes equations for weak entropy entropy flux pairs generated by compactly supported c2 test functions are confined in a compact set in h 1 which leads to the existence of measure valued solutions that are confined by the tartar murat commutator relation a careful characterization of the unbounded support of the measure valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a dirac mass which leads to the convergence of solutions of the navier stokes equations to a finite energy entropy solution of the isentropic euler equations with finite energy initial data relative to the different end states at infinity 2010 wiley periodicals inc

effective velocity in compressible navier stokes equations with third order derivatives abstract a formulation of certain barotropic compressible navier stokes equations with third order derivatives as a viscous euler system is proposed by using an effective velocity variable the equations model for instance viscous korteweg or quantum navier stokes flows the formulation in the new variable allows for the derivation of an entropy identity which is known as the bd bresch desjardins entropy equation as a consequence of this estimate a new global in time existence result for the one dimensional quantum navier stokes equations with strictly positive particle densities is proved

the nonrelativistic limit in radiation hydrodynamics i weak entropy solutions for a model problem abstract this paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions the system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integral type source term to a family of radiation transport equations we first show existence of entropy solutions to cauchy problems of the model system in the framework of functions of bounded variation this is done by using difference schemes and discrete ordinates then we establish strong convergence of the entropy solutions indexed with c as c goes to infinity the limit function satisfies a scalar integro differential equation

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

saint venant s problem and semi inverse solutions in nonlinear elasticity saint venant s problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross sectional resultants of force and moment using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite dimensional manifold in particular the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations moreover the set of semi inverse solutions can be analyzed locally

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

dynamics of singularity surfaces for compressible viscous flows in two space dimensions we prove the global existence of solutions of the navier stokes equations of compressible barotropic flow in two space dimensions which exhibit convecting singularity curves the fluid density and velocity gradient have jump discontinuities across these curves exactly as predicted by the rankine hugoniot conditions and these jump discontinuities decay exponentially in time more rapidly for smaller viscosities the singularity curves remain c1 despite the fact that the velocity fields which convect them are not continuously differentiable 2002 wiley periodicals inc

global classical solutions to the one dimensional compressible fluid models of korteweg type with large initial data abstract this paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the cauchy problem of the one dimensional isothermal compressible fluid models of korteweg type with density dependent viscosity coefficient and capillarity coefficient the case when the viscosity coefficient and the capillarity coefficient for some parameters r is considered under some conditions on we first show the global existence of large solutions around constant states if the far fields of the initial data are the same while if the far fields of the initial data are different we prove the global stability of rarefaction waves with large strength here global stability means the initial perturbation can be arbitrarily large our analysis is based on the elementary energy method and the technique developed by y kanel 29

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

navier stokes equations for compressible fluids global existence and qualitative properties of the solutions in the general case we consider the equations which describe the motion of a viscous compressible fluid taking into consideration the case of inflow and or outflow through the boundary by means of some a priori estimates we prove the existence of a global in time solution moreover as a consequence of a stability result we show that there exist a periodic solution and a stationary solution

global solutions of a high dimensional system for korteweg materials the global existence of solutions to a high dimensional system of korteweg materials is established when the initial data are small the complete model was proposed by dunn and serrin and the local existence of the solution was obtained in an earlier paper by the authors

quasi distribution semigroups and integrated semigroups abstract suppose x is a banach space we introduce the concept of quasi distribution semigroups on x as a generalization of the concept of distribution semigroups introduced by lions in 19 in our approach the generator a of a quasi distribution semigroup may not be densely defined also introduced is a functional calculus for a in terms of the fourier transform for fixed r 0 k n 0 define order r k for a quasi distribution semigroup and define an f r k functional calculus we prove that a generates a k 1 times integrated semigroup of exponential type r with local lipschitz continuity if and only if a generates a quasi distribution semigroup of order r k if and only if a has an f r k functional calculus when a is densely defined a quasi distribution semigroup reduces to a distribution semigroup in the sense of lions

numerical liapunov schmidt spectral methods for k determined problems we describe several generalized liapunov schmidt methods in the bordered systems formulation for operator equations and collect information concerning classification of bifurcation functions we present a short outline of spectral methods in this bifurcation context and show the convergence of the discrete defining equations and nondegeneracy conditions for spectral finite element and difference methods to the counterparts of the original operator for k determined problems this result is then applied to a model for turing patterns in a two component mixture

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

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

asymptotic stability of strong rarefaction waves for the compressible fluid models of korteweg type abstract this paper is concerned with the time asymptotic behavior toward strong rarefaction waves of solutions to one dimension compressible fluid models of korteweg type which governs the motions of the compressible fluids with internal capillarity assume that the corresponding riemann problem to the compressible euler system can be solved by rarefaction waves v r u r t x if the initial data is a small perturbation of an approximate rarefaction wave for v r u r t x we show that the corresponding cauchy problem admits a unique global smooth solution which tends to v r u r t x time asymptotically since we do not require the strength of the rarefaction waves to be small this result gives the nonlinear stability of strong rarefaction waves for the one dimensional compressible fluid models of korteweg type the analysis is based on the elementary l 2 energy method together with continuation argument

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

relative entropy in multi phase models of 1d elastodynamics convergence of a non local to a local model abstract in this paper we study a local and a non local regularization of the system of nonlinear elastodynamics with a non convex energy we show that solutions of the non local model converge to those of the local model in a certain regime the arguments are based on the relative entropy framework and provide an example of how local and non local regularizations may compensate for non convexity of the energy and enable the use of the relative entropy stability theory even if the energy is not quasi or poly convex

global existence for 1d compressible isentropic navier stokes equations with large initial data we prove the global existence of weak solutions of the cauchy problem for the navier stokes equations of compressible isentropic flow of a polytropic gas in one space dimension the initial velocity and density are assumed to be in l2 and l2 n bv respectively modulo additive constants in particular no smallness assumptions are made about the intial data in addition we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3 2

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

global solutions to the equations of a ginzburg landau theory for structural phase transitions in shape memory alloys abstract global existence of smooth solutions is proved for the system of partial differential equations governing the dynamics of martensic phase transitions in shape memory alloys the free energy is assumed in ginzburg landau form and nonconvex in the order parameter

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

the incompressible limit for compressible mhd equations in lp type critical spaces abstract in this paper the global well posedness and the low mach number limit for compressible viscous magnetohydrodynamic equations in the critical l p type besov space are considered more precisely we will show that in the isentropic case the solution of compressible magnetohydrodynamic equations will converge to that of incompressible magnetohydrodynamic equations when the mach number tends to zero in the critical l p framework moreover the convergence rate will be obtained

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

characterization and computation of singular points with maximum rank deficiency characterizations are given of points at which a mapping has a rank drop possibly greater than one multiple bifurcation points can then be computed without needing a full unfolding of the singularity nondegeneracy conditions are studied in detail

travelling waves for the gross pitaevskii equation ii the purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the gross pitaevskii equation in dimensions two and three our arguments based on minimization under constraints yield a full branch of solutions and extend earlier results see 3 4 8 where only a part of the branch was built in dimension three we also show that there are no travelling wave solutions of small energy

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

numerical solution of navier stokes korteweg systems by local discontinuous galerkin methods in multiple space dimensions compressible liquid vapor flow with phase transitions can be described by systems of navier stokes korteweg type they extend the navier stokes equations by nonlinear higher grade terms which take the form of either differential or nonlocal integral operators a numerical approximation method on the basis of the local discontinuous galerkin method in multiple space dimensions is suggested for isothermal flows it relies on a specific discretization of a non conservative formulation to enhance the performance of the overall scheme two techniques are used i local spatial adaptivity based on gradient indicators for the density and ii parallelism based on domain decomposition the paper concludes with numerical experiments in two and three space dimensions they show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

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

bordered augmented linear systems in numerical continuation and bifurcation we present the algorithm gmbe to solve a block linear system begin gathered begin array 20 c n m 1 end array left begin array 20 c a 11 0 a 31 end array begin array 20 c a 12 a 22 a 32 end array begin array 20 c a 13 a 23 a 33 end array right left begin array 20 c x 1 x 2 x 3 end array right left begin array 20 c f 1 f 2 f 3 end array right begin array 20 c n m 1 end array hfill nm111 hfill end gathered or m z h which appears in numerical computation of folds and symmetry breaking bifurcation gmbe principally uses 1 solve witha 11 t anda 22 t each and 2 solves witha 11 anda 22 each m must be well conditioned buta 11 anda 22 may be arbitrarily ill conditioned in fact singular to machine precision the error analysis requires that the solvers fora 11 a 22 a 11 t anda 22 t are stable in the sense of backward projection of the errors and that the solvers fora 11 anda 22 are bounded in a sense to be made precise both properties are typically possessed in practice by solvers based on either direct or iterative methods gmbe is the first algorithm that solvesmz h by using the solvers fora 11 etc as black boxes and does not require prior computation of the left and right singular vectors ofa 11 anda 22

existence of global strong solution for korteweg system with large infinite energy initial data this work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see cite fds which can be used as a phase transition model we will prove the existence of local and global under a condition of smallness on the initial data strong solutions with discontinuous initial density where ln rho 0 belongs to the besov space b n 2 infty r n in passing our result allow to consider initial data with discontinuous interfaces up our knowledge in the literature the results of existence of strong solutions require generally continuous initial density it implies also that we can work with initial data space of infinite energy our result relies on the fact that the density can be written as the sum of the solution rho l associated to linear system and a remainder density bar rho which is more regular than rho l by taking into account the regularizing effects induced on the bilinear convection term the main difficulty concerns the proof of new estimate of maximum principle type for the linear system associated to the korteweg system the proof is based on a characterization of the besov space in terms of the semi group associated to this linear system let also mention that we prove the existence of global strong solution with a smallness hypothesis which is subcritical in terms of the scaling of the equations it allows us to exhibit a family of large energy initial data for the scaling of the equations providing global strong solution in particular for the first time up our knowledge we show the existence of global strong solution for some large energy initial data when n 2 we finish this paper by introducing the notion of quasi solutions for the korteweg s system a tool which has been developed in the framework of the compressible navier stokes equations cite arxiv arxiv1 hal cras1 cras2 which enables us to improve the previous result and by obtaining the existence of global strong solution with large initial velocity in b n 1 2 infty in particular when n 2 it implies the existence of global strong solution with large energy data as a corollary we get global existence and uniqueness for highly compressiblekorteweg system when n geq2 it means that for any large initial data under an irrotational condition on the initial velocity we have the existence of global strong solution provided that the pressure is sufficiently highly compressible

local numerical analysis of hopf bifurcation this paper contributes to a local numerical analysis of hopf bifurcation in the sense of a bordered systems approach however the method proposed enables us to decide about stability exchange betw

scaling solution branches of one parameter bifurcation problems we consider an algorithm for analyzing bifurcation structure and for branch switching in solution branches of one parameter dependent problems based on liapunov schmidt methods and analyses of scales of solutions we verify the existence of bifurcating solution branches successively via truncated taylor expansions of the reduced bifurcation equations and an enlarged system as examples bifurcations of semilinear elliptic differential equations are studied on square and hexagon domains

the approximation of generalized turn ng points by projections methods with superconvergence to the critical parameter a procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between banach spaces this characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization applications are given to parameter dependent two point boundary value problems in particular collocation at gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

ill posed problems associated with sectorial operators in banach space abstract ill posed problems of the form u t a u h 0 t t u 0 are studied in banach space under two related scenarios both of which consider a well posed logarithmic approximation in the first scenario a generates a bounded holomorphic semigroup of angle 2 and so a is sectorial of angle 0 in the second where may be relaxed to any value in 0 2 we impose an additional assumption that e a is sectorial regularization results for the original ill posed problem are then established in the linear case where h h t and subsequently extended to nonlinear problems where h is replaced by h t u t we compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations both linear and nonlinear in l p r n 1 p

unbounded holomorphic functional calculus and abstract cauchy problems for operators with polynomially bounded resolvents abstract we consider linear unbounded operators b on a banach space x whose spectrum is contained in an open set v such that the resolvent w b 1 exists and is o 1 w for w v for some 1 we will call this an operator of type v for such operators we construct a functional calculus b defined for functions holomorphic on v whose real part is bounded above such that b generates an exponentially bounded b m semigroup where m 2 v this map is a continuous algebra homomorphism in a sense that is made precise later we give best possible spectral mapping theorems applications of this construction include the following we characterize subsets v of the complex plane with the property that whenever a is densely defined and of type v then the abstract cauchy problem has a solution for all initial data in a dense set we show that when b is of a type v then there exists a ranach subspace y such that d b m y x and there exists an h v functional calculus for b y on this interpolating space the imaginary powers of b form a c 0 group and we have spectral decompositions of b when v is disconnected our construction produces solutions for all initial data in dense sets of wide classes of first and second order cauchy problems including the schrodinger equation the backwards heat equation and the laplace equation on l p r n or l p i p where is a region in r n with smooth boundary we also consider incomplete cauchy problems of arbitrarily high order we emphasize that in our construction of b we do not assume that is meromorphic at this allows us to define exponentials cosines and logarithms of b directly for large classes of operators b for example we may define the exponential e 5 of an operator b with real spectrum

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

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

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

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 strong solution for the korteweg system with quantum pressure in dimension n ge 2 this work is devoted to prove the existence of global strong solution in dimension n ge 2 for a isothermal model of capillary fluids derived by dunn and serrin see arch ration mech anal 88 2 95 133 1985 which can be used as a phase transition model we will restrict us to the case of the so called compressible navier stokes system with quantum pressure which corresponds to the korteweg system with capillary coefficient such that kappa rho frac kappa 1 rho with kappa 1 0 and rho the density in a first part we prove the existence of strong solution in finite time for large initial data in critical besov spaces with precise estimate on the life span t the second part consists in proving the existence of global strong solution with large initial data for a specific choice on the capillary coefficient kappa 1 mu 2 with mu the viscosity coefficient to do this we derive different energy estimates on the density and the effective velocity v which allow us to extend the strong solution beyond t the main difficulty consists in estimating the l infty norm of frac 1 rho the proof relies mostly on a method introduced by de giorgi mem accad sci torino cl sci fis mat nat 3 3 25 43 1957 see also ladyzhenskaya et al linear and quasilinear equations of parabolic type ams translations providence 1968 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

optimal decay rate for the compressible navier stokes poisson system in the critical l p framework abstract in this paper we consider the large time behavior of global solutions to the initial value problem for the compressible navier stokes poisson system in the l p critical framework and in any dimension n 3 we obtain the time decay rates not only for lebesgue spaces but also for a family of besov norms with negative or nonnegative regularity exponents which improves the decay results in high sobolev regularity the proof is mainly based on the littlewood paley theory and refined time weighted inequalities in fourier space

lagrangean structure and propagation of singularities in multidimensional compressible flow we study the propagation of singularities in solutions of the navier stokes equations of compressible barotropic fluid flow in two and three space dimensions the solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded initial energies are small and initial velocities are in certain fractional sobolev spaces we show that if the initial density is bounded below away from zero in an open set v then each point of v determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi holder homeomorphism of v onto its image at each positive time this lagrangean structure is then applied to show that if the initial density has a limit at a point of such a set v from a given side of a continuous hypersurface in v then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface which is also a continuous manifold if the limits from both sides exist then the rankine hugoniot conditions hold in a strict pointwise sense showing that the jump in the divergence of the velocity is proportional to the jump in the pressure this leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density from which it follows that discontinuities persist for all time convecting along fluid particle paths and in the case that the pressure is strictly increasing in density having strengths which decay exponentially in time

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

on the global well posedness of the euler boussinesq system with fractional dissipation abstract we study the global well posedness of the euler boussinesq system with the term dissipation d on the temperature equation we prove that for 1 the coupled system has a global unique solution for initial data with critical regularities

global existence and optimal decay rate of the compressible navier stokes korteweg equations with external force abstract in this paper we consider the three dimensional compressible fluid models of korteweg type the equation takes the form of the compressible navier stokes equation with the cauchy stress tensor in the momentum equations under the smallness assumption of the external force in some sobolev space we first show the existence of stationary solution by standard iterative arguments next we present that the global in time solution of the initial value problem for the three dimensional compressible navier stokes korteweg equations exists uniquely and approaches the stationary solution as t provided the prescribed initial data and the external force are sufficiently small finally based on the elaborate energy estimates of the solution for the nonlinear system and l 2 decay estimates of semigroup generated by the corresponding linearized equation we show the optimal l 2 convergence rates of the solution towards the stationary solution as far as we know this is the first result about the global existence and l 2 decay rate of smooth solutions for the compressible navier stokes korteweg equations with the external force

optimal time decay estimates for the compressible navier stokes equations in the critical l p framework the global existence issue for the isentropic compressible navier stokes equations in the critical regularity framework was addressed in danchin invent math 141 3 579 614 2000 more than 15 years ago however whether optimal time decay rates could be shown in critical spaces has remained an open question here we give a positive answer to that issue not only in the l2 critical framework of danchin invent math 141 3 579 614 2000 but also in the general lp critical framework of charve and danchin arch ration mech anal 198 1 233 271 2010 chen et al commun pure appl math 63 9 1173 1224 2010 haspot arch ration mech anal 202 2 427 460 2011 we show that under a mild additional decay assumption that is satisfied if for example the low frequencies of the initial data are in l p 2 mathbb r d the lp norm the slightly stronger dot b 0 p 1 norm in fact of the critical global solutions decays like t d frac 1p frac14 for t to infty exactly as firstly observed by matsumura and nishida in proc jpn acad ser a 55 337 342 1979 in the case p 2 and d 3 for solutions with high sobolev regularity

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

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation

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

a numerical liapunov schmidt method with applications to hopf bifurcation on a square we discuss an iterative method for calculating the reduced bifurcation equation of the liapunov schmidt method and its numerical approximation using appropriate genericity assumptions with symmetry we derive a taylor series for the reduced equation where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation this method is used to calculate reduced equations at hopf bifurcation of the two dimensional brusselator equations on a square with neumann and dirichlet boundary conditions we examine several hopf bifurcations within the three parameter space there are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods

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

computation of takens bogdanov type bifurcations with arbitrary codimension let dot u f u alpha u 0 u 0 describe a dynamical system where u in r n is the state variable alpha in r k is a parameter variable and f u alpha takes values in r n here n can be large but k is usually small say leqslant 10 a solution u alpha to the equilibrium equations f u alpha 0 is a takens bogdanov type bifurcation of codimension m if the jacobian f u u alpha contains an m dimensional jordan block corresponding to a zero eigenvalue and has no other eigenvalues equal to zero this includes the cases of turning point bifurcation m 1 takens bogdanov bifurcation m 2 and triple point bifurcation m 3 a unified direct method is proposed based on ideas of a griewank and g w reddien in the case of takens bogdanov points for both the detection and computation of parameter values a at which thkens bogdanov type bifurcations of any given codimension occur as an application we compute tkens bogdanov type bifurcations with codi