h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

nonlinear noise excitation of intermittent stochastic pdes and the topology of lca groups consider the stochastic heat equation tu lu u where l denotes the generator of a levy process on a locally compact hausdor abelian group g r r is lipschitz continuous 1 is a large parameter and denotes space time white noise on r g the main result of this paper contains a near dichotomy for the expected squared energy e kutk 22 g of the solution roughly speaking that dichotomy says that in all known cases where u is intermittent the energy of the solution behaves generically as expfconst 2 g when g is discrete and expfconst 4 g when g is connected

intermittency for stochastic partial differential equations driven by strongly inhomogeneous space time white noises abstract in this paper the main topic is to investigate the intermittent property of the one dimensional stochastic heat equation driven by an inhomogeneous brownian sheet which is a noise deduced from the study of the catalytic super brownian motion under some proper conditions on the catalytic measure of the inhomogeneous brownian sheet we show that the solution is weakly full intermittent based on the estimates of moments of the solution in particular it is proved that the second moment of the solution grows at the exponential rate the novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the lebesgue measure on r

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

on the existence and position of the farthest peaks of a family of stochastic heat and wave equations we study the stochastic heat equation partial t u mathcal l u sigma u dot w in 1 1 dimensions where dot w is space time white noise r r is lipschitz continuous and mathcal l is the generator of a symmetric levy process that has finite exponential moments and u0 has exponential decay at we prove that under natural conditions on i the th absolute moment of the solution to our stochastic heat equation grows exponentially with time and ii the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting see however gartner et al in probab theory relat fields 111 17 55 1998 gartner et al in ann probab 35 439 499 2007 for the analysis of the location of the peaks in a different model finally we show that these results extend to the stochastic wave equation driven by laplacian

asymptotic properties of some space time fractional stochastic equations consider non linear time fractional stochastic heat type equations of the following type begin aligned partial beta tu t x nu delta alpha 2 u t x i 1 beta t lambda sigma u mathop f limits cdot t x end aligned in d 1 dimensions where nu 0 beta in 0 1 alpha in 0 2 the operator partial beta t is the caputo fractional derivative while delta alpha 2 is the generator of an isotropic stable process and i 1 beta t is the riesz fractional integral operator the forcing noise denoted by mathop f limits cdot t x is a gaussian noise and the multiplicative non linearity sigma mathbb r rightarrow mathbb r is assumed to be globally lipschitz continuous mijena and nane stochastic process appl 125 9 3301 3326 2015 have introduced these time fractional spdes these types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory under suitable conditions on the initial function we study the asymptotic behaviour of the solution with respect to time and the parameter lambda in particular our results are significant extensions of those in ann probab to appear foondun and khoshnevisan electron j probab 14 21 548 568 2009 mijena and nane 2015 and mijena and nane potential anal 44 295 312 2016 along the way we prove a number of interesting properties about the deterministic counterpart of the equation

moment bounds for a class of fractional stochastic heat equations we consider fractional stochastic heat equations of the form frac partial u t x partial t delta alpha 2 u t x lambda sigma u t x dot f t x here dot f denotes the noise term under suitable assumptions we show that the second moment of the solution grows exponentially with time in particular this answers an open problem in cite cokh along the way we prove a number of other interesting properties which extend and complement results in cite foonjose cite khoshnevisan 2013aa and cite khoshnevisan 2013ab

a delta well with a mass jump in this paper it is shown that a one dimensional hamiltonian with an attractive delta potential at the origin plus a mass jump at the same point cannot have a bound state as is the case with the ordinary attractive delta potential with constant mass unless another term is added into the potential in the form of a derivative of the dirac delta at the origin the consistency of this singular potential with two terms is guaranteed by choosing suitable matching conditions at the singular point for the wavefunctions furthermore it is proved that the self adjointness of the hamiltonian with both singular interactions determines the coefficient of the derivative of the delta in a unique manner under these conditions the bound state and its energy are obtained and it is checked that the correct results in the limit of equal masses are obtained

trilinear resonant interactions of semilinear hyperbolic waves an important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions or approximate solutions one of the difficulty is known to be the nonlinear interaction of oscillations this phenomenon has been studied for special classes of solutions such as oscillatory functions with phases and amplitudes this is the domain of geometric optics in some cases general families of bounded solutions have also been considered the present paper enters the later category but is highly inspired by the results of nonlinear geometric optics geometric optics provides a precise description of the asymptotic structure of solutions to both linear e g wh and nonlinear j hk mr hmr jmr 1 jmr 2 s hyperbolic pdes whose initial data u 0 x u0 x eu1 x 0 x e have rapid oscillations with prescribed phases 0 and a single prescribed scale 1e such solutions are important in both theory and applications l kk in the lowest approximation oscillations having different spatial scales interact only when m equals zero since otherwise the amplitudes of higher frequency oscillations become negligible relative to those of lower frequencies taking m 0 requires l estimates systems that are well posed in l occur mostly in one spatial dimension for such systems arbitrary uniformly bounded sequences of initial data may be considered but comparatively little is known about the asymptotic structure of the resulting solutions since the oscillations in such solutions may be extremely complex a natural problem is to determine their non oscillatory part which may be identified with the weak limit of the sequence of solutions the highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general of course some information about those oscillations will generally be needed in order to determine the weak limit the asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity for entropy solutions of a genuinely nonlinear conservation law t 1 or of a pair of such equations in one spatial dimension dp oscillations of amplitude o 1 do not persist for positive times so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data although oscillations do persist in linear systems the weak limit of a sequence of solutions to such

on the strong pre compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux abstract under some non degeneracy condition we show that sequences of entropy solutions of a semi linear elliptic equation are strongly pre compact in the general case of a caratheodory flux vector the proofs are based on localization principles for h measures corresponding to sequences of measure valued functions

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

a wigner measure analysis of the dufort frankel scheme for the schrodinger equation we apply wigner transform techniques to the analysis of the dufort frankel difference scheme for the schrodinger equation and to the continuous analogue of the scheme in the case of a small scaled planck constant semiclassical regime in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop numerical test examples are presented to help interpret the theory and to compare the dufort frankel scheme to other difference schemes for the schr odinger equation

semiclassical completely integrable systems long time dynamics and observability via two microlocal wigner measures we look at the long time behavior of solutions to a semi classical schr odinger equation on the torus we consider time scales which go to infinity when the semi classical parameter goes to zero and we associate with each time scale the set of semi classical measures associated with all possible choices of initial data on each classical invariant torus the structure of semi classical measures is described in terms of two microlocal measures obeying explicit propagation laws we apply this construction in two directions we first analyze the regularity of semi classical measures and we emphasize the existence of a threshold for time scales below this threshold the set of semi classical measures contains measures which are singular with respect to lebesgue measure in the position variable while at and beyond the threshold all the semi classical measures are absolutely continuous in the position variable reflecting the dispersive properties of the equation second the techniques of two microlocal analysis introduced in the paper are used to prove semiclassical observability estimates the results apply as well to general quantum completely integrable systems

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

optimal bounds and microgeometries for elastic two phase composites consider two isotropic linearly elastic materials with elasticity tensors c 1 c 2 and assume c 1 c 2 we prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite rank laminates this implies that for any imposed uniform strain or stress field optimal bounds on the effective strain or stress energy per unit volume are attained by finite rank laminates explicit bounds on the strain energy with no symmetry assumptions are given in dimensions 2 and 3 these bounds are more stringent than the classical voigt reuss bounds finally explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry

on the calculation of energy minimizing phase fractions in shape memory alloys modeling of multi phase materials such as shape memory alloys usually leads to non quasi convex energies such formulations give rise to the development of microstructure and are therefore not suitable for immediate computation of the material behavior to circumvent this problem many models make use of the convex reus lower bound as an estimate of the relaxed free energy to evaluate the quality of this estimate upper bounds have been constructed by lamination in recent literature showing a small difference of upper and lower bounds in many examples in this work we compare upper bounds by lamination and lower bounds by convexification regarding the energy optimal phase fractions as well as the globally minimized energies and the overall stresses resulting from the two approaches similarities and differences of the results obtained with both bounds are discussed

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

global compensated compactness theorem for general differential operators of first order let a1 x d and a2 x d be differential operators of the first order acting on l vector functions u u 1 ldots u l in a bounded domain omega subset mathbb r n with the smooth boundary partial omega we assume that the h1 norm u h 1 omega is equivalent to sum i 1 2 a iu l 2 omega b 1u h frac 1 2 partial omega and sum i 1 2 a iu l 2 omega b 2u h frac 1 2 partial omega where bi bi x is the trace operator onto partial omega associated with ai x d for i 1 2 which is determined by the stokes integral formula unit outer normal to partial omega furthermore we impose on a1 and a2 a cancellation property such as a 1a 2 prime 0 and a 2a 1 prime 0 where a prime i is the formal adjoint differential operator of ai i 1 2 suppose that u m m 1 infty and v m m 1 infty converge to u and v weakly in l 2 omega respectively assume also that a 1 u m m 1 infty and a 2 v m m 1 infty are bounded in l 2 omega if either b 1 u m m 1 infty or b 2 v m m 1 infty is bounded in h frac 1 2 partial omega then it holds that int omega u m cdot v m rm d x to int omega u cdot v rm d x we also discuss a corresponding result on compact riemannian manifolds with boundary

the phase flow method this paper introduces the phase flow method a novel accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations the method operates by initially constructing the phase map for small times using a standard ode integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow the computational complexity of building up the complete phase map is usually that of tracing a few rays in addition the phase flow method is provably and empirically very accurate once the phase map is available integrating the ode for initial conditions on the invariant manifold only makes use of local interpolation thus having constant complexity the paper develops applications in the field of high frequency wave propagation and shows how to use the phase flow method to 1 rapidly propagate wave fronts 2 rapidly calculate wave amplitudes along these wave fronts and 3 rapidly evaluate multiple wave arrival times at arbitrary locations

on the hormander mihlin theorem for mixed norm lebesgue spaces abstract for some questions in the theory of partial differential equations it is useful to consider the mixed norm lebesgue spaces introduced by benedek and panzone 1962 we show that hormander s results 1960 obtained on translation invariant operators on lebesgue spaces extend to the mixed norm case we also revisit the hormander mihlin multiplier theorem in this setting which was first obtained by lizorkin 1970 where we particularly consider the form of the constants used in the estimates this improvement allows us to prove the boundedness of classical pseudodifferential operators on mixed norm lebesgue spaces as well as generalising the h distributions a recently introduced 2011 by mitrovic and the first author extension of h measures to these spaces

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

erratum to existence and strong pre compactness properties for entropy solutions of a first order quasilinear equation with discontinuous flux sequences of entropy solutions of a non degenerate first order quasilinear equation are shown to be strongly pre compact in the general case of a caratheodory flux vector existence of the weak and entropy solution to the cauchy problem for such an evolutionary equation is also established the proofs are based on the general localization principle for h measures corresponding to sequences of measure valued functions