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

on the spreading of characteristics for non convex conservation laws we study the spreading of characteristics for a class of one dimensional scalar conservation laws for which the ux function has one point of in ection it is well known that in the convex case the characteristic speed satises a one sided lipschitz estimate using dafermos theory of generalized characteristics we show that the characteristic speed in the non convex case satises an holder estimate in addition we give a one sided lipschitz estimate with an error term given by the decrease of the total variation of the solution

lax oleĭnik explicit formula and structure theory for balance laws abstract in this paper we study the structure of solutions for the balance laws in one space dimension with a source term having the form u t f u x t u 0 in r 0 u x 0 u 0 on r t 0 consequently we show that the behaviour of the shock packets and the phenomenon of one shock situation is completely different when compared to equations without a source term moreover contrary to the behaviour of shock packets in conservation laws in our situation we show that the slope of the left boundary of the shock packet can be strictly greater than that of the right boundary

rings and modules which are stable under automorphisms of their injective hulls it is proved among other results that a prime right nonsingular ring in particular a simple ring r is right self injective if rr is invariant under automorphisms of its injective hull this answers two questions raised by singh and srivastava and clark and huynh an example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption also shown is that automorphism invariant modules are precisely pseudo injective modules answering a recent question of lee and zhou furthermore rings whose cyclic modules are automorphism invariant are investigated

rigorous derivation of nonlinear scalar conservation laws from follow the leader type models via many particle limit we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow the leader type model which is interpreted as the discrete lagrangian approximation of the nonlinear scalar conservation law more precisely we prove that the empirical measure respectively the discretised density obtained from the follow the leader system converges in the 1 wasserstein topology respectively in mathbf l 1 loc to the unique kru kov entropy solution of the conservation law the initial data are taken in mathbf l infty nonnegative and with compact support hence we are able to handle densities with a vacuum our result holds for a reasonably general class of velocity maps including all the relevant examples in the applications for example in the lighthill whitham richards model for traffic flow with a possible degenerate slope near the vacuum state the proof of the result is based on discrete mathbf bv estimates and on a discrete version of the one sided oleinik type condition in particular we prove that the regularizing effect mathbf l infty mapsto mathbf bv for nonlinear scalar conservation laws is intrinsic to the discrete model

convergence results for the flux identification in a scalar conservation law here we study an inverse problem for a quasilinear hyperbolic equation we start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function then we use a lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation despite the fact that the lagrangian formulation is formal and that the cost function is not necessarily differentiable a viscous perturbation and a numerical approximation of the problem allow us to justify this computation when the adjoint problem for the quasi linear equation admits a smooth solution then the perturbed adjoint states can be proved to converge to that very solution the sequences of gradients for both perturbed problems are also proved to converge to the same element of the subdifferential of the cost function we evidence these results for a large class of numerical schemes and particular cost functions which can be applied to the identification of isotherms for conservation laws modeling distillation or chromatography they are illustrated by numerical examples

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

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

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

modules which are coinvariant under automorphisms of their projective covers abstract in this paper we study modules coinvariant under automorphisms of their projective covers we first provide an alternative and in fact a more succinct and conceptual proof for the result that a module m is invariant under automorphisms of its injective envelope if and only if given any submodule n of m any monomorphism f n m can be extended to an endomorphism of m and then as a dual of it we show that over a right perfect ring a module m is coinvariant under automorphisms of its projective cover if and only if for every submodule n of m any epimorphism m m n can be lifted to an endomorphism of m

automorphism invariant modules satisfy the exchange property abstract warfield proved that every injective module has the exchange property this was generalized by fuchs who showed that quasi injective modules satisfy the exchange property we extend this further and prove that a module invariant under automorphisms of its injective hull satisfies the exchange property we also show that automorphism invariant modules are clean and that directly finite automorphism invariant modules satisfy the internal cancellation and hence the cancellation property

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

weak approximation for cubic hypersurfaces we prove weak approximation for smooth cubic hypersurfaces of dimension at least 2 defined over the function field of a complex curve

weak approximation for isotrivial families we study weak approximation for isotrivial families of rationally connected varieties over a smooth projective curve in the case where the family is formed by taking a quotient of a cyclic group action we find a necessary and sufficient condition for weak approximation to hold in most other cases there is also a sufficient condition we also check that this condition is satisfied for all smooth fano hypersurfaces in projective spaces

the sharp form of oleĭnik s entropy condition in several space variables we investigate the conditions under which the volpert kruzkov solution of a single conservation law in several space variables with flux f will satisfy the simplified entropy condition div f u s l t and when this condition guarantees uniqueness for given lx cauchy data we show that when f is c1 our condition guarantees uniqueness iff f is isotropic and that for such f the volpert kruzkov solution always satisfies our condition

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

long time stability in systems of conservation laws using relative entropy energy we study the long time stability of shock free solutions of hyperbolic systems of conservation laws under an arbitrarily large initial disturbance in l2 l we use the relative entropy method a robust tool which allows us to consider rough and large disturbances we display practical examples in several space dimensions for scalar equations as well as isentropic gas dynamics for full gas dynamics we use a trick from chen 1 in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy

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

on the attainable set for a scalar nonconvex conservation law we deal with a cauchy problem for a scalar conservation law in one space dimension the flux function is assumed to be nonconvex in particular to have a single inflection point we consider a compactly supported initial datum and regard it as a control the main result of the paper states sufficient conditions for a function v to be attained at a fixed time t by a trajectory of the conservation law

continuous modules are clean ar ingr is said to be clean if every element of r is a sum of an idempotent and a unit the class of clean rings is quite large and includes for instance semiperfect rings and thus finite rings and rings of linear transformations of vector spaces we prove that the endomorphism ring of every continuous or discrete module is clean

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

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

adjoint based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws we propose a rigorous procedure to obtain the adjoint based gradient representation of cost functionals for the optimal control of discontinuous solutions of conservation laws hereby it is not necessary to introduce adjoint variables for the shock positions our approach is based on stability properties of the adjoint equation we give a complete analysis for the case of convex scalar conservation laws the adjoint equation is a transport equation with discontinuous coefficients and special reversible solutions must be considered to obtain the correct adjoint based gradient formula reversible solutions of the adjoint transport equation and the required stability properties are analyzed in detail

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

dual automorphism invariant modules a module m is called an automorphism invariant module if every isomorphism between two essential submodules of m extends to an automor phism of m this paper introduces the notion of dual of such modules we call a module m to be a dual automorphism invariant module if whenever k1 and k2 are small submodules of m then any epimorphism m k1 m k2 with small kernel lifts to an endomorphism of m in this paper we give vari ous examples of dual automorphism invariant module and study its properties in particular we study abelian groups and prove that dual automorphism invariant abelian groups must be reduced it is shown that over a right perfect ring r a lifting right r module m is dual automorphism invariant if and only if m is quasi projective all our rings have identity element and modules are right unital a right r module m is called an automorphism invariant module if every isomorphism between two essential submodules of m extends to an automorphism of m equivalently m is an automorphism invariant module if for any automorphismofe m m m where e m is the injective hull of m see 6 and 10

existence and uniqueness of solutions for a non uniformly parabolic equation abstract in this paper we study a generalized burgers equation ut u2 x tuxx which is a non uniformly parabolic equation for t 0 we show the existence and uniqueness of classical solutions to the initial value problem of the generalized burgers equation with rough initial data belonging to l r

a sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms we present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term the sensitivity analysis is based on shift variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations based on a first order approximation by shift variations in l1 we introduce the concept of shift differentiability which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking type functionals in the main part of the paper we show that entropy solutions are generically shift differentiable at almost all times t 0 with respect to the control hereby we admit shift variations for the initial data which allows us to use the shift differentiability result repeatedly over time slabs this is useful for the design of optimization methods with time domain decomposition our analysis especially of the shock sensitivity combines structural results by using generalized characteristics and an adjoint argument our adjoint based shock sensitivity analysis does not require us to restrict the richness of the shock structure a priori and admits shock generation points the analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one sided lipschitz condition as a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking type functionals

existence and decay rates of solutions to the generalized burgers equation in this paper we study the generalized burgers equation ut u 2 2 x f t uxx w here f t 0 for t 0 we show the existence and uniqueness of classical solutions to the initial value problem of the generalized burgers equation with rough initial data belonging to l r as well it is obtained the decay rates of u in l p norm are algebra order for p 1

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

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

total oscillation diminishing property for scalar conservation laws we prove a bv estimate for scalar conservation laws that generalizes the classical total variation diminishing property in fact for any lipschitz continuous monotone we have that u tv is nonincreasing in time we call this property total oscillation diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on tvd schemes we also show that semi discrete total variation diminishing finite volume schemes are tod and that the fully discrete godunov scheme is tod

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