a nonexistence result to a cauchy problem in nonlinear one dimensional thermoelasticity considering the cauchy problem for a nonlinear system arising in thermoelasticity it is proved that its solution develops singularities in finite time depending on the size and regularity of the initial data the present work is distinguished from a previous work by relaxing the requirements on the initial data and allowing for a slightly more general and nonautonomous forcing term besides permitting the insertion of gradient terms in both equations of the system

global existence and exponential stability in one dimensional nonlinear thermoelasticity with thermal memory in this paper we study the global existence uniqueness and asymptotic behaviour of solutions to the equations of one dimensional nonlinear thermoelasticity with thermal memory subject to dirichlet dirichlet boundary conditions for the temperature di5erence and displacement respectively the reference con6guration under consideration is the unit interval 0 1 the equations under consideration read as follows

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

strong solutions to the cauchy problem of two dimensional incompressible fluid models of korteweg type abstract this paper studies the local existence of strong solutions to the cauchy problem of the incompressible fluid models of korteweg type with vacuum as far field density the corresponding 3d problem has been solved by tan and wang 2010 21 notice that the technique used by tan and wang fails treating the 2d case because the l p norm p 2 of the velocity u cannot be controlled in terms only of u and u here in the present paper we will use the framework of weighted approximation estimates introduced by liang 2015 14 for navier stokes equations to obtain the local existence of strong solutions provided the initial density does not decay very slowly at infinity with the compact support case included

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

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

global strong solutions for the incompressible nematic liquid crystal flows with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible nematic liquid crystal flows with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in vacuum provided u 0 l 2 d 0 l 2 is suitably small with arbitrary large initial density which extended the local strong solution by gao et al 12 to be a global one

evaluating infinite integrals involving products of bessel functions of arbitrary order the difficulties involved with evaluating infinite integrals involving products of bessel functions are considered and a method for evaluating these integrals is outlined the method makes use of extrapolation on a sequence of partial sums and requires rewriting the product of bessel functions as the sum of two more well behaved functions numerical results are presented to demonstrate the efficiency of this method where it is shown to be significantly superior to standard infinite integration routines

mildly dissipative nonlinear timoshenko systems global existence and exponential stability we consider nonlinear systems of timoshenko type in a one dimensional bounded domain the system has a dissipative mechanism being present only in the equation for the rotation angle it is a damping effect through heat conduction the global existence of small smooth solutions as well as the exponential stability are investigated

global solutions of the neumann problem in one dimensional nonlinear thermoelasticity for the equations of one dimensional nonlinear thermoelasticity slemrod 17 established in 1981 the global existence of smooth solutions for small initial data in the case of a bounded reference configuration for a homogeneous medium the boundary conditions traction free constant temperature or rigidly clamped insulated he considered yield through the differential equation an additional boundary condition which was crucial for obtaining the necessary energy estimates subsequently jiang 7 discussed the same boundary conditions as in 17 in the case of an unbounded reference configuration the half line kawashima lo zheng and shen 181 and hrusa and tarabek 6 dealt with the one dimensional cauchy problem recently racke and shibata 13 proved a global existence theorem for the dirichlet initial boundary value problem rigidly clamped constant temperature in a bounded interval by using the decay estimates for the linearized equations and the l energy method and the author 8 obtained a similar result for the problem in an unbounded interval the half line by only applying the l energy method more recently using a method similar to that in 13 shibata 16 showed that global small solutions exist for the neumann problem traction free insulated in the case of a bounded reference configuration for an unbounded reference configuration the half line the case of the neumann boundary conditions remained open our purpose in this paper is to prove a global existence theorem for the neumann problem in the half line and in a bounded interval for small data by applying the l2 energy method the proof given here closely follows the pattern of the proof in 8 subject to the necessary modifications we note that a globally defined classical solution should not be expected for large data indeed dafermos and hsiao 4 and hrusa and messaoudi 5 showed that for specialized constitutive equations if the initial data are large then the solution to the cauchy problem will blow up in finite time for three dimensional nonlinear models we refer the reader to the papers 3 9 10 12 151 and the references cited therein this paper is organized as follows in section 1 the equations of one dimensional nonlinear thermoelasticity are presented and the main theorem is stated in section 2 we give a local

blow up of solutions to the cauchy problem in nonlinear one dimensional thermoelasticity this paper is concerned with the blow up phenomena of solutions to the cauchy problem in non autonomous nonlinear one dimensional thermoelastic models obeying both fourier s law of heat flux and the theory due to gurtin and pipkin moreover some previously related results have been extended

global smooth solutions and asymptotic stability in one dimensional nonlinear thermoelasticity the global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one dimensional homogeneous reference configuration dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t is described

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

global 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

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

existence and exponential decay in nonlinear thermoelasticity it is well known that in the absence of dissipation smooth solution of nonlinear elastic materials develop singularities in finite time while for thermoelastic materials the conduction of the heat equation provides dissipation that competes with the destabilizing effect of nonlinearity in the elastic response the level of subtlety of this dissipation depends on the boundary condition that the displacement and the thermal difference are supporting slemrod i showed the global existence of smooth solution for small data when the boundary is either traction free and at a constant temperature or rigidly clamped and thermally insulated a similar result was obtained by zheng 2 these boundary conditions get a simpler damping mechanism because they imply additional boundary conditions for u and the thermal difference 8 that is if an end is clamped then the displacement u and the thermal difference 0 satisfy u 0 and 0 0 there respectively so we can make additional partial integrations which led to the desire a priori l2 estimate in case of dirichlet boundary condition for which the boundary is rigidly clamped and held at a constant temperature we lost the value of u in that point and instead of it we get u of 0 so this case leads ill behaved boundary terms and it is not possible to apply directly the multiplicative techniques to secure global estimate recently racke and shibata 3 proved global existence of a smooth solution for these boundary conditions to do this the authors showed the algebraic decay of the energy for the linear equation by studying the spectral properties of the stationary linearized problem the rate of decay depends on higher regularity of the initial data and therefore the global existence result depends on the initial data to be small in hm o l with m large one of the authors of this paper proved in 4 see also the work of kim 5 that the solution of the linearized thermoelastic system decays exponentially as time goes to infinity this fact allows us to get simpler existence result for the corresponding nonlinear equation as was shown in 6 for small data uo ul in h3 0 l x h2 0 l the system in question is written as follows

global solutions to the 3 d incompressible inhomogeneous navier stokes system in this paper we consider the global well posedness of the 3 d incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces a0 bq 13q r3 u0 u0h u03 bp 1 1 3p r3 for p q satisfying 1 q p 6 and 1q 1p 13 more precisely we prove that there exist two positive constants c0 c0 such that if a0 bq 13q u0h bp 1 1 3p exp c0 u03 bp 1 1 3p2 2 c0 then 1 3 has a unique global solution a l r bq 13q r3 u l r bp 1 1 3p r3 l1 r bp 11 3p r3 in particular this result implies the global well posedness result in abidi and paicu 2007 2 for the inhomogeneous navier stokes system with small initial data

exponential energy decay in a linear thermoelastic rod abstract we examine the stability of solutions to a pair of coupled linear partial differential equations which describe the temperature distribution and displacement within a one dimensional thermoelastic rod for particular sets of natural boundary conditions the eigenfunctions are shown to form a riesz basis on the hilbert space of finite energy states for these cases asymptotic eigenvalue estimates show that the spectrum is to the left of some vertical line re contained in the open left half of the complex plane an energy decay estimate of the form e t me t e 0 is then obtained

general boundary stabilization of memory type thermoelasticity in this paper we consider an n dimensional system of thermoelasticity where a viscoelastic dissipation is acting on a part of the boundary by using a lemma first appeared in the work of messaoudi and soufyane general decay of solutions of a wave equation with a boundary control of memory type nonlinear anal real world appl to be published we establish a general decay result from which the usual exponential and polynomial decay are only special cases our result improves earlier ones in the literature and extends the general result of messaoudi and soufyane general decay of solutions of a wave equation with a boundary control of memory type nonlinear anal real world appl 11 2896 2010 to thermoelastic systems

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

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