one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

dynamics in the fundamental solution of a non convex conservation law there is a huge jump in the theory of conservation laws if the convexity assumption is dropped in this paper we study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution three extra shock types are introduced other than the usual genuine shock which are left right and double sided contacts there are three kinds of phenomena of these shocks which are called branching merging and transforming all of these shocks and phenomena can be observed if the flux function has two inflection points a comprehensive picture of a global dynamics of a nonconvex flux is discussed in terms of characteristic maps and dynamical convex concave envelopes

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

coding into ramsey sets in 6 w t gowers formulated and proved a ramsey type result which lies at the heart of his famous dichotomy for banach spaces he defines the notion of weakly ramsey set of block sequences of an infinite dimensional banach space and shows that every analytic set of block sequences is weakly ramsey we show here that gowers result follows quite directly from the fact that all g sets are weakly ramsey if the banach space does not contain c0 and from the fact that all f sets are weakly ramsey in the case of an arbitrary banach space we also show that every result obtained by the application of gowers theorem to an analytic set can also be obtained by applying the theorem to a f set or to a g set if the space does not contain c0 this fact explains why the only known applications of this technique are based on very low ranked borel sets open closed f or g

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

bv solutions for the cauchy problem of a quasilinear hyperbolic equation with σ finite borel measure and nonlinear source the aim of this paper is to discuss the cauchy problem of the quasilinear hyperbolic equation of the form ut um x up with nonnegative finite borel measures as initial conditions where m 1 0 p 1 are some given real numbers in particular the existence of bv solutions for the above problem is obtained

an infinite ramsey theorem and some banach space dichotomies a problem of banach asks whether every infinite dimensional banach space which is isomorphic to all its infinite dimensional subspaces must be isomorphic to a separable hilbert space in this paper we prove a result of a ramsey theoretic nature which implies an interesting dichotomy for subspaces of banach spaces combined with a result of komorowski and tomczakjaegermann this gives a positive answer to banachxs problem we then generalize the ramsey theoretic result and deduce a further dichotomy for banach spaces with an unconditional basis

a non diagonalizable pure state we construct a pure state on the c algebra formula see text of all bounded linear operators on formula see text which is not diagonalizable i e it is not of the form formula see text for any orthonormal basis formula see text of formula see text and an ultrafilter u on n this constitutes a counterexample to anderson s conjecture without additional hypothesis and improves results of c akemann n weaver i farah and i smythe who constructed such states making additional set theoretic assumptions it follows from results of j anderson and the positive solution to the kadison singer problem due to a marcus d spielman and n srivastava that the restriction of our pure state to any atomic masa formula see text of diagonal operators with respect to an orthonormal basis formula see text is not multiplicative on formula see text

boundary behavior of the fast diffusion equation the fast diffusion equation avm vt 0 m 1 is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established in this paper we are going to obtain a lipschitz growth rate of the solution at the boundary of nl and estimate that in terms of the various data

very singular solution and short time asymptotic behaviors of nonnegative singular solutions for heat equation with nonlinear convection abstract in this paper we study the following cauchy problem u t u x x u n x x t r 0 u x 0 0 x 0 where parameter n 0 its nonnegative solution is called singular solution when u x t satisfies the equation in the sense of distribution initial conditions in the classical sense and also u x t exhibits a singularity at the origin 0 0 as we know the singular solution is called source type solution if the initial is m x where x is dirac measure and constant m 0 the solution is called very singular solution if it possesses more singularity than that of source type solution at the origin here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process we find critical values n 2 n 1 n 0 such that there exists unique source type solution in the exponent range of 0 n n 0 while there exists no nonnegative singular solution if n n 0 only in the case of n 2 n n 1 there exists a very singular solution but in the case of n n 1 or n n 2 there is no solution that exhibits more singular than source type solution at the origin furthermore we describe the short time asymptotic behavior of the singular solutions when such cauchy problem is solvability for source type solution or very singular solution

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

three dimensional reductions of four dimensional quasilinear systems in this paper we show that four dimensional quasilinear systems of first order integrable by the method of two dimensional hydrodynamic reductions possess infinitely many three dimensional hydrodynamic reductions which are also integrable systems these three dimensional multi component integrable systems are irreducible to two dimensional hydrodynamic reductions in a generic case

existence and uniqueness of bv solutions for a conservation law withσ finite borel measures as initial conditions abstract the aim of this paper is to discuss the existence and uniqueness of the cauchy problem for a conservation law of the form u t u m x 0 m 1 with finite borel measures as initial conditions in particular we also obtain a non existence theorem which implies that our results are optimal

ergodic banach spaces abstract we show that any banach space contains a continuum of non isomorphic subspaces or a minimal subspace we define an ergodic banach space x as a space such that e 0 borel reduces to isomorphism on the set of subspaces of x and show that every banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block subspaces we also use our methods to get uniformity results we show that an unconditional basis of a banach space of which every block subspace is complemented must be asymptotically c 0 or l p and we deduce some new characterisations of the classical spaces c 0 and l p

asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to n waves abstract the goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws general convex flux f u 0 is considered with an assumption lim u 0 uf u f u 1 we show that under suitable conditions on the initial value its solution converges to an n wave in l1 norm with the optimal convergence order of o 1 t the technique we use in this article is to enclose the solution with two rarefaction waves we also show a uniform convergence order in the sense of graphs a numerical example of this phenomenon is included

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

behaviour of the velocity of one dimensional flows in porous media the one dimensional flow of gas of density u through a porous medium obeys the equation u um x where m 1 x e r and t 0 we prove that the local velocity of the gas given by v mu 2u1 not only is bounded for t t 0 but approaches an n wave profile as t oo n waves are the typical asymptotic profiles for some first order conservation laws a class of nonlinear hiperbolic equations the case m 0 0 2 u x 0 u0 x forx e r where m is a positive constant and u0 satisfies 0 3 u0 e l r u0 0 a e u0 0 apart from its mathematical interest as a simple model of nonlinear evolution equation of possibly degenerate or singular parabolic type equation 0 1 and its n dimensional analogue appear in a number of applications cf 18 for a reference in the typical application to the flow of gas through a porous medium we have m 1 and u stands for the density of the gas writing 0 1 as a conservation law for the mass

asymptotics in the critical case for whitham type equations abstract consider the cauchy problem for nonlinear dissipative evolution equations u t n u u l u 0 x r t 0 u 0 x u 0 x x r where l is the linear pseudodifferential operator l u f x l u and the nonlinearity is a quadratic pseudodifferential operator n u u f x r a t y u t y u t y d y u f x u is direct fourier transformation let the initial data u 0 h 0 h 0 1 2 are sufficiently small and have a non zero total mass m u 0 x d x 0 here h n m l 2 x m i x n x l 2 is the weighted sobolev space then we prove that the main term of the large time asymptotics of solutions in the critical case is given by the self similar solution defined uniquely by the total mass m of the initial data

b h has a pure state that is not multiplicative on any masa assuming the continuum hypothesis we prove that h has a pure state whose restriction to any masa is not pure this resolves negatively old conjectures of kadison and singer and of anderson

asymptotic estimates of solutions ofut 12δu u in r rd d 2 abstract we express the solution u of u t 1 2 u u in r r d d 2 u 0 through a functional of a d dimensional bessel process where is invariant by rotations and profiled this allows us to estimate the asymptotic behaviour of u t 1 and u t as t goes to infinity

algebraic varieties in the birkhoff strata of the grassmannian gr 2 harrison cohomology and integrable systems the local properties of the families of algebraic subsets wg in the birkhoff strata 2g of gr 2 containing the hyperelliptic curves of genus g are studied it is shown that the tangent spaces tg for wg are isomorphic to the linear spaces of 2 coboundaries particular subsets in wg are described by the integrable dispersionless coupled kdv systems of hydrodynamical type defining a special class of 2 cocycles and 2 coboundaries in tg it is demonstrated that the blows ups of such 2 cocycles and 2 coboundaries and gradient catastrophes for associated integrable systems are interrelated

hodograph solutions of the dispersionless coupled kdv hierarchies critical points and the euler poisson darboux equation it is shown that the hodograph solutions of the dispersionless coupled kdv dckdv hierarchies describe critical and degenerate critical points of a scalar function which obeys the euler poisson darboux equation singular sectors of each dckdv hierarchy are found to be described by solutions of higher genus dckdv hierarchies concrete solutions exhibiting shock type singularities are presented

on the blow up of solutions of a convective reaction diffusion equation we study the blow up of positive solutions of the cauchy problem for the semilinear parabolic equation where u is a scalar function of the spatial variable x n and time t 0 a v a 0 1 p and 1 q we show that a if p 1 and 1 q p there always exist solutions which blow up in finite time b if 1 q p min 1 2 n 1 2 q n 1 or if q 1 and 1 p l 2 n then all positive solutions blow up in finite time c if q 1 and p min 1 2 n 1 2 q n 1 then global solutions exist d if q 1 and p 1 2 n then global solutions exist

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

n waves in elasticity we study the large time behavior of solutions of the one dimensional equations of elasticity typically a nonconvex system of conservation laws we show that solutions with compact initial support converge in l1 to a superposition of n waves at algebraic rate likewise we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order our analysis uses approximate scalar laws together with l1 stability of scalar solutions and decay in total variation of the wave speed

existence and uniqueness for a quasilinear hyperbolic equation with σ finite borel measures as initial conditions abstract the aim of this paper is to discuss the cauchy problem for a quasilinear hyperbolic equation of the form u t u m x u p 0 m 1 p 0 with finite borel measures as initial conditions in particular the existence and uniqueness of global bv solutions with any nonnegative finite borel measures as initial conditions are obtained in the case m p m 1

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

coisotropic deformations of algebraic varieties and integrable systems coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a poisson ideal it is shown that coisotropic deformations of sets of intersection points of plane quadrics cubics and space algebraic curves are governed in particular by the dkp wdvv dvn d2dtl equations and other integrable hydrodynamical type systems particular attention is paid to the study of two and three dimensional deformations of elliptic curves the problem of an appropriate choice of the poisson structure is discussed

determinacy and weakly ramsey sets in banach spaces we give a sufficient condition for a set of block subspaces in an infinite dimensional banach space to be weakly ramsey using this condition we prove that in the levy collapse of a mahlo cardinal every projective set is weakly ramsey this together with a construction of w h woodin is used to show that the axiom of projective determinacy implies that every projective set is weakly ramsey in the case of c 0 we prove similar results for a stronger ramsey property and for hereditarily indecomposable spaces we show that the axiom of determinacy plus the axiom of dependent choices imply that every set is weakly ramsey these results are the generalizations to the class of projective sets of some theorems from w t gowers and our paper weakly ramsey sets in banach spaces

quasiclassical limit of coupled kdv equations riemann invariants and multi hamiltonian structure abstract we consider the quasiclassical limit of the first nontrivial flow in coupled kdv hierarchy written down in riemann invariants it assumes the extremely simple form r i k 1 n r k 2r i r x i i 1 n up to certain natural equivalence this turns out to be the only hydrodynamic system with n 1 compatible purely local hamiltonian structures

asymptotic behavior of solutions of a conservation law without convexity conditions in this paper we study the asymptotic behavior for the hyperbolic conservation law u i 0 t 0 oo x co 1 1 u x 0 uo x al x co 1 2 with riemann like data for x large the function f is a smooth nonlinear function of u in general eq 1 1 does not have a continuous solution for all time shock curves appear after finite time we will consider a piecewise continuous weak solution of 1 1 19 lo it is well known that across a discontinuity line x x t the solution satisfies the rankine hugoniot condifion r h and the entropy condition e 111

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

long time behaviour for a nonlinear first order equation abstract in this paper we study the asymptotic behaviour of solutions to the cauchy problem for the nonlinear balance law u t u m x u m 0 with m 1 and bounded nonnegative compactly supported initial data we prove the convergence with a precise rate to a fundamental solution we also obtain a precise description of the interface

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

piecewise self similar solutions and a numerical scheme for scalar conservation laws the solution of the riemann problem was a building block for general cauchy problems in conservation laws a cauchy problem is approximated by a series of riemann problems in many numerical schemes but since the structure of the riemann solution holds locally in time only and furthermore a riemann solution is not piecewise constant in general there are several fundamental issues in this approach such as the stability and the complexity of computation in this article we introduce a new approach which is based on piecewise self similar solutions the scheme proposed in this article solves the problem without the time marching process the approximation error enters in the step for the initial discretization only which is given as a similarity summation of base functions the complexity of the scheme is linear convergence to the entropy solution and the error estimate are shown the mechanism of the scheme is introduced in detail together with several interesting properties of the scheme

weakly ramsey sets in banach spaces gowers analysis of the combinatorial content of his celebrated dichotomy for infinite dimensional separable banach spaces 7 led him to the formulation of the property of being weakly ramsey applied to sets of block bases a combinatorial notion related to the classical ramsey property for infinite sets of positive integers let n be the set of all infinite sets of positive integers with the natural topology induced by the cantor space via characteristic functions n is a polish space a subset n is ramsey if there exists a n such that either a or a where a is the set of all infinite subsets of a the famous galvin prikry theorem asserts that every borel subset of n is ramsey 4 or equivalently that every borel map from n into a finite space is constant on a cube a silver 17 shows that in fact all analytic subsets of n i e the continuous images of borel sets are ramsey a simpler more combinatorial proof is given by ellentuck 3 the ramsey property for more complex subsets of n turns out to depend essentially on the axioms of set theory thus for instance while go del s axiom of constructibility implies that some continuous image of a co analytic set is not ramsey see 9 martin s axiom implies that all such sets are ramsey 17 furthermore large cardinal axioms or determinacy axioms imply that all projective or doi 10 1006 aima 2001 1983 available online at http www idealibrary com on

on the upper semicontinuity of a quasiconcave functional abstract in the recent paper 21 the second author proved a divergence quasiconcavity inequality for the following functional d a t n det a x 1 n 1 dx defined on the space of positive definite matrices in l p t n sym n with zero divergence we consider the space x p of tensor fields in l p t n sym n whose divergence is a radon measure we endow x p with the weak topology given by the weak convergence in l p and the weak convergence of the measures representing the divergence of the tensor fields our main result proves the weak upper semicontinuity of the functional d on x p if and only if p n n 1 we also consider the case p n n 1 and show that d is upper semicontinuous along sequences satisfying additional conditions we use the positive result to show some properties of multi dimensional burgers equation

flat bidifferential ideals and semi hamiltonian pdes in this paper we consider a class of semi hamiltonian systems characterized by the existence of a special conservation law the density and the current of this conservation law satisfy a second order system of pdes which has a natural interpretation in the theory of flat bidifferential ideals the class of systems we consider contains important well known examples of semi hamiltonian systems other examples such as genus 1 whitham modulation equations for kdv are related to this class by a reciprocal transformation

potential comparison and asymptotics in scalar conservation laws without convexity abstract two kinds of optimal convergence orders in l 1 norm to a self similar solution are proved or conjectured for various evolutionary problems so far the first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1 t employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials we present the technique clearly in the simple setting of scalar conservation laws in one space dimension

souslin partitions of products of finite sets abstract to every infinite sequence of positive integers m m i i we associate two fields of sets a field cl m of subsets of and a field pcl m of subsets of their relevance to ramsey theory is based on the fact that for every cl m measurable partition c 2 there is a sequence h i i with hi mi such that c is constant on i hi similarly for every pcl m measurable partition c 2 there is h and a sequence h i i of sets with hi and hi mi such that c is constant on i hi h in di prisco et al j combin theory ser a 93 2001 333 combinatorica to appear it is shown that cl m and pcl m are fields that contain all closed sets and therefore all borel subsets of their corresponding domains we show here that they are in fact closed under souslin operation and that under suitable assumptions they contain all reasonably definable subsets of their corresponding domains these results are then used to show that the classical partition relation is not equivalent to its polarized version solving thus a long standing problem in this area see j symbolic logic 58 1998 860 notas de logica matematica vol 39 universidad nacional del sur bahia blanca argentina 1994 pp 89 94

regularizing effects for ut aϑ u 0 in l1 abstract various initial boundary value problems and cauchy problems can be written in the form du dt a u 0 where r r is nondecreasing and a is the linear generator of strongly continuous nonexpansive semigroup e ta in an l1 space for example if a subject perhaps to suitable boundary conditions we obtain equations arising in flow in a porous medium or plasma physics depending on the choice of while if a x acting in l1 r we have a scalar conservation law in this paper we show that if m m 0 and m 2 m 2 where 1 1 then roughly speaking the norm of t du dt may be estimated in terms of the initial data u0 in l1 such estimates give information about the regularity of solutions asymptotic behaviour etc in applications side issues such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator a are also dealt with these considerations are of independent interest

singular sector of the burgers hopf hierarchy and deformations of hyperelliptic curves we discuss the structure of shock singularities of the burgers hopf hierarchy it is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy the stratification is associated with the birkhoff decomposition of the grassmannian given by the set of linear spaces spanned by the hierarchy we then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum the hierarchy is called the hidden burgers hopf hierarchy and we found the riemann invariant form and the hodograph solution

contractivity of wasserstein metrics and asymptotic profiles for scalar conservation laws abstract the aim of this paper is to analyze contractivity properties of wasserstein type metrics for one dimensional scalar conservation laws with nonnegative l and compactly supported initial data and its implications on the long time asymptotics the flux is assumed to be convex and without any growth condition at the zero state we propose a time parameterized family of functions as intermediate asymptotics and prove the solutions after a time depending scaling converge toward this family in the d wasserstein metric this asymptotic behavior relies on the aforementioned contraction property for conservation laws in the space of probability densities metrized with the d wasserstein distance finally we also give asymptotic profiles for initial data whose distributional derivative is a probability measure

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

on strongly summable ultrafilters and union ultrafilters we prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered union ultrafilters are not we also prove that the existence of ultrafilters of these sorts implies the existence of p points and therefore cannot be established in zfc