boundary effect on asymptotic behaviour of solutions to the p system with linear damping we consider the asymptotic behaviour of solutions to the p system with linear damping on the half line r 0 vt ux 0 ut p v x u with the dirichlet boundary condition u x 0 0 or the neumann boundary condition ux x 0 0 the initial date v0 u0 x has the constant state v u at x l hsiao and t p liu commun math phys 143 1992 599 605 have shown that the solution to the corresponding cauchy problem behaves like diffusion wave and k nishihara j differential equations131 1996 171 188 137 1997 384 395 has proved its optimal convergence rate our main concern in this paper is the boundary effect in the case of null dirichlet boundary condition on u the solution v u is proved to tend to v 0 as t tends to infinity its optimal convergence rate is also obtained by using the green function of the diffusion equation with constant coefficients in the case of null neumann boundary condition on u v 0 t is conservative and v 0 t v0 0 by virtue of the first equation so that v x t is expected to tend to the diffusion wave v x t connecting v0 0 and v in fact the solution v u x t is proved to tend to v x t 0 in the special case v0 0 v the optimal convergence rate is also obtained however this is not known in the case v0 0 v

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

convergence of the fractional step lax friedrichs scheme and godunov scheme for the isentropic system of gas dynamics a convergence theorem of the fractional step lax friedrichs scheme and godunov scheme for an inhomogeneous system of isentropic gas dynamics 1 5 3 is established by using the framework of compensated compactness meanwhile a corresponding existence theorem of global solutions with large data containing the vacuum is obtained

l 1 convergence to the barenblatt solution for compressible euler equations with damping we study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite the model system is the compressible euler equation with frictional damping as t the density is conjectured to obey the well known porous medium equation and the momentum is expected to be formulated by darcy s law in this paper we prove that any l weak entropy solution to the cauchy problem of damped euler equations with finite initial mass converges strongly in the natural l1 topology with decay rates to the barenblatt profile of the porous medium equation the density function tends to the barenblatt solution of the porous medium equation while the momentum is described by darcy s law the results are achieved through a comprehensive entropy analysis capturing the dissipative character of the problem

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

symmetry breaking at non positive solutions of semilinear elliptic equations we consider symmetry breaking bifurcations at non positive radially symmetric solutions of semilinear elliptic equations on a ball with dirichlet boundary conditions for nonlinearities which are asymptotically affine linear we find solutions at which the symmetry breaks the kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group o n for this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough thus we obtain by the equivariant branching lemma a large variety of isotropy subgroups of o n which occur as symmetries of the bifurcating solution branches

l1 convergence rates to the barenblatt solution for the damped compressible euler equations abstract in our previous work 17 it is shown that any l weak entropy solution of damped compressible euler equation converges to the barrenblatt s solution with finite mass in l 1 norm with a convergence rate for 1 3 in this paper by introducing an elaborate iterative method and using the intensive entropy analysis we prove the l 1 convergence to the barrenblatt s solution for any large 3 furthermore we get better l 1 convergence rate for 2 3

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

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

butterflies ii torsors for 2 group stacks abstract we study torsors over 2 groups and their morphisms in particular we study the first non abelian cohomology group with values in a 2 group butterfly diagrams encode morphisms of 2 groups and we employ them to examine the functorial behavior of non abelian cohomology under change of coefficients we re interpret the first non abelian cohomology with coefficients in a 2 group in terms of gerbes bound by a crossed module our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the fraction weak morphism determined by a butterfly as a practical byproduct we show how butterflies can be used to obtain explicit maps at the cocycle level in addition we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2 groups as well as specific features pertaining to group extensions

extension theory and the calculus of butterflies abstract this paper provides a unified treatment of two distinct viewpoints concerning the classification of group extensions the first uses weak monoidal functors the second classifies extensions by means of suitable h 2 actions we develop our theory formally by making explicit a connection between non abelian g torsors and fibrations then we apply our general framework to the classification of extensions in a semi abelian context by means of butterflies 1 between internal crossed modules as a main result we get an internal version of dedecker s theorem on the classification of extensions of a group by a crossed module in the semi abelian context bourn s intrinsic schreier mac lane extension theorem 13 turns out to be an instance of our theorem 6 3 actually even just in the case of groups our approach reveals a result slightly more general than classical schreier mac lane theorem

biextensions of picard stacks and their homological interpretation abstract let s be a site we introduce the 2 category of biextensions of strictly commutative picard s stacks we define the pull back the push down and the sum of such biextensions and we compute their homological interpretation if p q and g are strictly commutative picard s stacks the equivalence classes of biextensions of p q by g are parametrized by the cohomology group ext 1 p l q g the isomorphism classes of arrows from such a biextension to itself are parametrized by the cohomology group ext 0 p l q g and the automorphisms of an arrow from such a biextension to itself are parametrized by the cohomology group ext 1 p l q g where p q and g are the complexes associated to p q and g respectively

hyperbolic to parabolic relaxation theory for quasilinear first order systems abstract in this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type in particular we obtain a complete theory for the 2 2 systems of genuinely nonlinear hyperbolic balance laws in 1 d with a strong dissipative term a different method which combines the div curl lemma with accretive operators is then applied to study the limiting profiles in the case of nonhomogeneous isentropic gas dynamics we also investigate relaxation results for some 2 d cases which include the cattaneo model for nonlinear heat conduction and the compressible euler flow moreover convergence result is also obtained for general semilinear systems in 1 d

group actions on algebraic stacks via butterflies abstract we introduce an explicit method for studying actions of a group stack g on an algebraic stack x as an example we study in detail the case where x p n 0 n r is a weighted projective stack over an arbitrary base s to this end we give an explicit description of the group stack of automorphisms of p n 0 n r the weighted projective general linear 2 group pgl n 0 n r as an application we use a result of colliot thelene to show that for every linear algebraic group g over an arbitrary base field k assumed to be reductive if char k 0 such that pic g 0 every action of g on p n 0 n r lifts to a linear action of g on a r 1

the snail lemma for internal groupoids abstract we establish a generalized form both of the gabriel zisman exact sequence associated with a pointed functor between pointed groupoids and of the brown exact sequence associated with a fibration of pointed groupoids our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers

lp convergence rate to nonlinear diffusion waves for p system with damping abstract in this paper we study the p system with frictional damping and show that the solutions time asymptotically tend to the nonlinear diffusion waves governed by the classical darcy s law by introducing an approximate green function we obtain the optimal l p 2 p convergence rate of the solution which is a perturbation of the nonlinear diffusion wave to the hyperbolic system

convergence to the barenblatt solution for the compressible euler equations with damping and vacuum we study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite the model system is the compressible euler equation with frictional damping as t the density is conjectured to obey the well known porous medium equation and the momentum is expected to be formulated by darcy s law in this paper we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data we prove that any l weak entropy solution to the cauchy problem of damped euler equations with finite initial mass converges strongly in lp with decay rates to matching barenblatt s profile of the porous medium equation the density function tends to the barenblatt s solution of the porous medium equation while the momentum is described by darcy s law

convergence to strong nonlinear diffusion waves for solutions of p system with damping this paper is concerned with the p system with frictional damping and our main purpose is two fold first we show that for a certain class of given large initial data nu 0 x u 0 x the cauchy problem 1 1 1 2 admits a unique global smooth solution nu t x u t x and such a solution tends time asymptotically at the optimal l p 2 less than or equal to p less than or equal to infinity decay rates to the corresponding nonlinear diffusion wave over bar t x u over bar t x governed by the classical darcy s law provided that the corresponding prescribed initial error function v 0 x u 0 x ties in h 3 x h 2 r boolean and l 1 x l 1 r compared with former results in this direction obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 k nishihara 1996 j differential equations 131 171 188 and k nishihara w k wang and t yang 2000 j differential equations 161 191 218 our main novelty lies in the facts that the nonlinear diffusion wave over bar t x u over bar t x need not to be weak and v 0 x u 0 x can be chosen arbitrarily large secondly we show that the nonlinear diffusion waves over bar t x u over bar t x are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance v 0 x u 0 x satisfies the assumption that parallel tov 0xx x parallel to l infinity parallel tou 0x x parallel to l infinity is sufficiently small we also show that the smallness assumption imposed oil the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 the smallness conditions we imposed on the initial disturbance are much more weaker c 2001 academic press

asymptotic behavior of the solutions to the damped compressible euler equations with vacuum abstract the asymptotic behavior of solutions of the damped compressible euler equations is conjectured to obey to the famous porous media equations pmes the previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic in present paper we prove that the l entropy weak solution with vacuum obtained by the compensated compactness theory converges strongly in l loc p 1 p space to the unique similarity solution of the related pme as time goes to infinity

convergence rate for compressible euler equations with damping and vacuum we study the asymptotic behavior of l weak entropy solutions to the compressible euler equations with damping and vacuum previous works on this topic are mainly concerned with the case away from the vacuum and small initial data in the present paper we prove that the entropy weak solution strongly converges to the similarity solution of the porous media equations in lp r 2 p with decay rates the initial data can contain vacuum and can be arbitrary large a new approach is introduced to control the singularity near vacuum for the desired estimates

compressible euler equations with vacuum abstract in this paper we first show that the regular solutions of compressible euler equations in r 3 with damping will not be global if the initial density function has compact support this implies that c 2 cannot be smooth across the boundary separating the gas and the vacuum after a finite time where c is the speed of sound then we study the local existence of solutions for isentropic gas flow in r when c 0 1 is smooth across using the energy method and the characteristics method

preservation of depth in the local geometric langlands correspondence it is expected that under mild conditions the local langlands correspondence preserves depths of representations in this article we formulate a conjectural geometrisation of this expectation we prove half of this conjecture by showing that the depth of a categorical representation of the loop group is greater than or equal to the depth of its underlying geometric langlands parameter a key ingredient of our proof is a new definition of the slope of a meromorphic connection a definition which uses opers

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

length 3 complexes of abelian sheaves and picard 2 stacks abstract we define a tricategory t 2 0 of length 3 complexes of abelian sheaves whose hom bigroupoids consist of weak morphisms of such complexes we also define a 3 category 2 pic s of picard 2 stacks whose hom 2 groupoids consist of additive 2 functors we prove that these categories are triequivalent as tricategories as a consequence we obtain a generalization of deligne s analogous result about picard stacks in sga4 expose xviii deligne 1973 11

butterflies i morphisms of 2 group stacks abstract weak morphisms of non abelian complexes of length 2 or crossed modules are morphisms of the associated 2 group stacks or gr stacks we present a full description of the weak morphisms in terms of diagrams we call butterflies we give a complete description of the resulting bicategory of crossed modules which we show is fibered and biequivalent to the 2 stack of 2 group stacks as a consequence we obtain a complete characterization of the non abelian derived category of complexes of length 2 deligne s analogous theorem in the case of picard stacks and abelian sheaves becomes an immediate corollary commutativity laws on 2 group stacks are also analyzed in terms of butterflies yielding new characterizations of braided symmetric and picard 2 group stacks furthermore the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non abelian cohomology removing a preexisting fibration condition on the coefficients short exact sequence

adjusted parallel transport for higher gauge theories many physical theories including notably string theory require non abelian higher gauge fields defining higher holonomy previous approaches to such higher connections on categorified principal bundles require these to be fake flat this condition however renders them locally gauge equivalent to connections on abelian gerbes for particular higher gauge groups for example 2 group models of the string group this limitation can be overcome by generalizing the notion of higher connection starting from this observation we define a corresponding generalized higher holonomy functor which is free from the fake flatness condition leading to a truly non abelian parallel transport

butterflies in a semi abelian context it is known that monoidal functors between internal groupoids in the category grp of groups constitute the bicategory of fractions of the 2 category grpd grp of internal groupoids internal functors and internal natural transformations in grp with respect to weak equivalences that is internal functors which are internally fully faithful and essentially surjective on objects monoidal functors can be equivalently described by a kind of weak morphisms introduced by b noohi under the name of butterflies in order to internalize monoidal functors in a wide context we introduce the notion of internal butterflies between internal crossed modules in a semi abelian category c and we show that they are morphisms of a bicategory b c our main result states that when in c the notions of huq commutator and smith commutator coincide then the bicategory b c of internal butterflies is the bicategory of fractions of grpd c with respect to weak equivalences

differential geometry of gerbes abstract we define in a global manner the notion of a connective structure for a gerbe on a space x when the gerbe is endowed with trivializing data with respect to an open cover of x we describe this connective structure in two separate ways which extend from abelian to general gerbes the corresponding descriptions due to j l brylinski and n hitchin we give a global definition of the 3 curvature of this connective structure as a 3 form on x with values in the lie stack of the gauge stack of the gerbe we also study this notion locally in terms of more traditional lie algebra valued 3 forms the bianchi identity which the curvature of a connection on a principal bundle satisfies is replaced here by a more elaborate equation