weak strong uniqueness for a bi fluid model for a mixture of non interacting compressible fluids abstract we investigate a version of one velocity baer nunziato type system with dissipation describing the motion of a mixture of two compressible fluids we define for this system weak solutions on one hand and dissipative weak solutions on the other hand and recall the theorem about their existence on a large time interval we investigate strong solutions and show their existence on a short time interval finally we prove that any weak solution satisfies a relative energy inequality and prove for this system the weak strong uniqueness principle this is the main result of the paper

the φ harmonic approximation and the regularity of φ harmonic maps abstract we extend the p harmonic approximation lemma proved by duzaar and mingione for p harmonic functions to harmonic functions where is a convex function the proof is direct and is based on the lipschitz truncation technique we apply the approximation lemma to prove holder continuity for the gradient of a solution of a harmonic system with critical growth

travelling wave solutions for a generalized double dispersion equation abstract we analyze the generalization of a double dispersion equation by means of the theory of symmetry reductions of partial differential equations we consider travelling wave reductions depending on the form of an arbitrary function we present some explicit solutions soliton solutions kinks and anti kinks

optimal l p l q estimates for parabolic boundary value problems with inhomogeneous data in this paper we investigate vector valued parabolic initial boundary value problems mathcal a t x d mathcal b j t x d subject to general boundary conditions in domains g in mathbb r n with compact c 2m boundary the top order coefficients of mathcal a are assumed to be continuous we characterize optimal l p l q regularity for the solution of such problems in terms of the data we also prove that the normal ellipticity condition on mathcal a and the lopatinskii shapiro condition on mathcal a mathcal b 1 dots mathcal b m are necessary for these l p l q estimates as a byproduct of the techniques being introduced we obtain new trace and extension results for sobolev spaces of mixed order and a characterization of triebel lizorkin spaces by boundary data

existence and blow up of solution of cauchy problem for the generalized damped multidimensional boussinesq equation we consider the existence both locally and globally in time and the blow up of solutions for the cauchy problem of the generalized damped multidimensional boussinesq equation

nonlinear elliptic systems with general growth we prove local lipschitz continuity and as a consequence c k and c regularity of weak solutions u for a class of nonlinear elliptic differential systems of the formn 1 xi a i du 0 1 2 m the growth conditions on the dependence of functions a i on the gradient du are so mild to allow us to embrace growths between the linear and the exponential cases and they are more general than those known in the literature

local higher integrability for unsteady motion equations of generalized newtonian fluids abstract in this paper we show local higher integrability of the gradient of weak solutions for unsteady motions of generalized newtonian fluids including electrorheological fluids under the restriction p z 3 n 2 n 2 moreover this is a new result even for corresponding stokes setting or a system with p structure depending on the symmetric gradient under the restriction p z 2 n n 2

riesz potentials and nonlinear parabolic equations the spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric riesz potential of the right hand side datum exactly as in the case of the heat equation heat kernels type estimates persist in the nonlinear case

linear convergence of an adaptive finite element method for the p laplacian equation we study an adaptive finite element method for the p laplacian like pdes using piecewise linear continuous functions the error is measured by means of the quasi norm of barrett and liu we provide residual based error estimators without a gap between the upper and lower bound we show linear convergence of the algorithm which is similar to the one of morin nochetto and siebert all results are obtained without extra marking for the oscillation

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

regularity for parabolic systems of uhlenbeck type with orlicz growth abstract we study the local regularity of p caloric functions or more generally of caloric functions in particular we study local solutions of non linear parabolic systems with homogeneous right hand side where the leading terms have uhlenbeck structure of orlicz type this paper closes the gap of 23 where liebermann proved that if the gradient of a solution is bounded it is holder continuous the crucial step is a novel local estimates for the gradient of the solutions which generalize and improve the pioneering estimates of dibenedetto and friedman 12 10 for the p laplace heat equation

cauchy problem of the generalized double dispersion equation abstract in this paper the existence and the uniqueness of the global solution for the cauchy problem of the generalized double dispersion equation are proved the blow up of the solution for the cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

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

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

holder zygmund estimates for degenerate parabolic systems abstract we consider energy solutions of the inhomogeneous parabolic p laplacian system t u div d u p 2 d u div g we show in the case p 2 that if the right hand side g is locally in l bmo then u is locally in l c 1 where c 1 is the 1 holder zygmund space this is the borderline case of the calderon zygmund theory we provide local quantitative estimates we also show that finer properties of g are conserved by du e g holder continuity moreover we prove a new decay for gradients of p caloric solutions for all 2 n n 2 p

existence of weak solutions for compressible navier stokes equations with entropy transport we consider the compressible navier stokes system with variable entropy the pressure is a nonlinear function of the density and the entropy potential temperature which unlike in the navier stokes fourier system satisfies only the transport equation we provide existence results within three alternative weak formulations of the corresponding classical problem our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory

regularity of solutions of nonlinear degenerate parabolic systems soit le systeme parabolique u i t div u p 2 u i 0 i m avec p 1 ou u i u i x t grad x et x r n ouvert pour toute solution de ce systeme avec max 1 2n n 2 p u est continu dans t

weak solutions for some compressible multicomponent fluid models the principle purpose of this work is to investigate a viscous version of a simple but still realistic bi fluid model described in bresch et al in giga novotn eds handbook of mathematical analysis in mechanics of viscous fluids 2018 whose non viscous version is derived from physical considerations in ishii and hibiki thermo fluid dynamics of two phase flow springer berlin 2006 as a particular sample of a multifluid model with algebraic closure the goal is to show the existence of weak solutions for large initial data on an arbitrarily large time interval we achieve this goal by transforming the model to a transformed two densities system which resembles the compressible navier stokes equations with however two continuity equations and a momentum equation endowed with the pressure of a complicated structure dependent on two variable densities the new transformed two densities system is then solved by an adaptation of the lions feireisl approach for solving compressible navier stokes equation completed with several observations related to the diperna lions transport theory inspired by maltese et al j differ equ 261 4448 4485 2016 and vasseur et al j math pures appl 125 247 282 2019 we also explain how these techniques can be generalized to a model of mixtures with more than two species this is the first result on the existence of weak solutions for any realistic multifluid system

a general regularity theorem for functionals with φ growth abstract we prove c 1 regularity for local minimizers of functionals u d x where is a young function in order to generalize the results of fuchs 2011 9 and diening et al 2009 6 we assume for the function only the 2 condition and t t t for a positive constant and of course a holder condition for the second derivatives