l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

l2 contraction of large planar shock waves for multi dimensional scalar viscous conservation laws abstract we consider a l 2 contraction a l 2 type stability of large viscous shock waves for the multi dimensional scalar viscous conservation laws up to a suitable shift by using the relative entropy methods quite different from the previous results we find a new way to determine the shift function which depends both on the time and space variables and solves a viscous hamilton jacobi type equation with source terms moreover we do not impose any conditions on the anti derivative variables of the perturbation around the shock profile more precisely it is proved that if the initial perturbation around the viscous shock wave is suitably small in l 2 norm then the l 2 contraction holds true for the viscous shock wave up to a suitable shift function note that bv norm or the l norm of the initial perturbation and the shock wave strength can be arbitrarily large furthermore as the time t tends to infinity the l 2 contraction holds true up to a spatially homogeneous time dependent shift function in particular if we choose some special initial perturbations then l 2 convergence of the solutions towards the associated shock profile can be proved up to a time dependent shift

l 2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension and we investigate the behavior of bounded l2 perturbations of shock wave solutions to the riemann problem using the relative entropy method we show that up to a time dependent translation of the shock the l2 norm of a perturbed solution relative to the shock wave is bounded above by the l2 norm of the initial perturbation

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

lattices for landau ginzburg orbifolds we consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries berglund hubsch and henningson proposed a duality between such pairs giving rise to mirror symmetry we define an orbifoldized signature for such a pair using the orbifoldized elliptic genus in the case of three variables and based on the homological mirror symmetry picture we introduce two integral lattices a transcendental and an algebraic one we show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature finally we give some evidence that these lattices are interchanged under the duality of pairs

relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non bv perturbations we develop a theory based on relative entropy to show the uniqueness and l 2 stability up to a translation of extremal entropic rankine hugoniot discontinuities for systems of conservation laws typically 1 shocks n shocks 1 contact discontinuities and n contact discontinuities of large amplitude among bounded entropic weak solutions having an additional trace property the existence of a convex entropy is needed no bv estimate is needed on the weak solutions considered the theory holds without smallness conditions the assumptions are quite general for instance strict hyperbolicity is not needed globally for fluid mechanics the theory handles solutions with vacuums

no contractive metrics for systems of conservation laws soit u t f u x 0 un systeme 2 2 de lois de conservation satisfaisant certaines hypotheses generiques sur f dans un voisinage 7b n de l espace u on montre que pour chaque metrique non degeneree d sur l u espace il existe des etats u 1 et u 2 dans 7b n tels que d u x t u 1 dx est une fonction strictement croissante de t dans un voisinage de t 0 ou u est la solution admissible du systeme avec u x 0 u 1 x 0 u 2 0 x 1 u 1 x 1

strong traces for solutions of multidimensional scalar conservation laws in this paper we consider multidimensional scalar conservation laws without bv estimates defined in a subset d we show that with a non degeneracy hypothesis on the flux we can define a strong notion of trace at the boundary of reached by l 1 convergence

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

a holder continuous ode related to traffic flow this paper is devoted to the proof of the well posedness of a class of ordinary differential equations odes the vector field depends on the solution to a scalar conservation law forward uniqueness of filippov solutions is obtained as well as their holder continuous dependence on the initial data of the ode furthermore we prove the continuous dependence in c 0 of the solution to the ode from the initial data of the conservation law in l 1 this problem is motivated by a model of traffic flow

on the dynamical rayleigh taylor instability the classical rayleigh taylor instability is shown to be valid in a nonlinear dynamical sense for smooth density profiles

uniqueness and stability of riemann solutions with large oscillation in gas dynamics we prove the uniqueness of riemann solutions in the class of entropy solutions in with arbitrarily large oscillation for the 3 3 system of euler equations in gas dynamics the proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane the uniqueness of riemann solutions yields their inviscid large time stability under arbitrarily large perturbation of the riemann initial data as long as the corresponding solutions are in l and have local bounded total variation satisfying a natural condition on its growth with time no specific reference to any particular method for constructing the entropy solutions is needed the uniqueness result for riemann solutions can easily be extended to entropy solutions u x t piecewise lipschitz in x for any t 0 with arbitrarily large oscillation

triangulated categories of matrix factorizations for regular systems of weights with e 1 abstract we construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a nondegenerate regular system of weights whose smallest exponents are equal to 1 in the associated grothendieck group the strongly exceptional collection defines a root basis of a generalized root system of sign l 0 2 and a coxeter element of finite order whose primitive eigenvector is a regular element in the expanded symmetric domain of type iv with respect to the weyl group

criteria on contractions for entropic discontinuities of systems of conservation laws we study the contraction properties up to shift for admissible rankine hugoniot discontinuities of n times n systems of conservation laws endowed with a convex entropy we first generalize the criterion developed in serre and vasseur j l ecole polytech 1 2014 using the spatially inhomogeneous pseudo distance introduced in vasseur contemp math ams 2013 our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems including the euler system moreover we introduce necessary conditions for contraction specifically targeted for intermediate shocks as an application we show that intermediate shocks of the two dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties we also investigate the contraction properties for contact discontinuities of the euler system for a certain range of contraction weights none of the results involve any smallness condition on the initial perturbation or on the size of the shock

long time asymptotics of a multiscale model for polymeric fluid flows in this paper we investigate the long time behavior of some micro macro models for polymeric fluids hookean model and fene model in various settings shear flow general bounded domain with homogeneous dirichlet boundary conditions on the velocity general bounded domain with non homogeneous dirichlet boundary conditions on the velocity we use both probabilistic approaches coupling methods and analytic approaches entropy methods

weak strong uniqueness for the isothermal navier stokes equations in this paper we are concerned with weak strong uniqueness results for the isothermal navier stokes equations in two space dimension using the methods of relative entropy we obtain some conditions on a weak solution such as the ones built up by plotnikov and weigant siam j math anal 47 626 653 2015 so that it is unique

measure valued solutions of scalar conservation laws with boundary conditions we define a solution concept for measure valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions this result may be used to prove convergence towards the unique solution for approximate solutions which are uniformly bounded in l weakly consistent with certain entropy inequalities and strongly consistent with the initial condition i e without using derivative estimates as an example convergence of a finite element method is demonstrated

nonlinear instability of hele shaw flows with smooth viscous profiles abstract we rigorously derive nonlinear instability of hele shaw flows moving with a constant velocity in the presence of smooth viscosity profiles where the viscosity upstream is lower than the viscosity downstream this is a single layer problem without any material interface the instability of the basic flow is driven by a viscosity gradient as opposed to conventional interfacial saffman taylor instability where the instability is driven by a viscosity jump across the interface existing analytical techniques are used in this paper to establish nonlinear instability

instability of periodic bgk equilibria a collisionless plasma is described by the vlasov poisson equations the bgk equilibria were proposed in 1957 as the simplest spatially dependent equilibria since that time the question of their instability has been an important open problem we prove that certain periodic bgk equilibria are unstable 1995 john wiley sons inc

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

motivic integration over deligne mumford stacks abstract the aim of this article is to develop the theory of motivic integration over deligne mumford stacks and to apply it to the birational geometry of deligne mumford stacks

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

boundary conditions for scalar conservation laws from a kinetic point of view boundary conditions for multidimensional scalar conservation laws are obtained in the context of hydrodynamic limits from a kinetic point of view the initial boundary value kinetic problem is well posed since inward and outward characteristics of the domain can be distinguished the convergence of the first momentum of the distribution function to an entropy solution of the conservation law is established boundary conditions are obtained the equivalence with the bardos leroux and nedelec conditions is studied

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

lp stability for entropy solutions of scalar conservation laws with strict convex flux abstract here we consider the scalar convex conservation laws in one space dimension with strictly convex flux which is in c 1 existence uniqueness and l 1 contractivity were proved by kru kov 14 using the relative entropy method leger showed that for a uniformly convex flux and for the shock wave solutions the l 2 norm of a perturbed solution relative to the shock wave is bounded by the l 2 norm of the initial perturbation here we generalize the result to l p norm for all 1 p also we show that for the non shock wave solution l p norm of the perturbed solution relative to the modified n wave is bounded by the l p norm of the initial perturbation for all 1 p

matrix factorizations and representations of quivers ii type ade case abstract we study a triangulated category of graded matrix factorizations for a polynomial of type ade we show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding dynkin quiver also we discuss a special stability condition for the triangulated category in the sense of t bridgeland which is naturally defined by the grading