the cauchy problem for indefinite improper affine spheres and their hessian equation abstract we give a conformal representation for indefinite improper affine spheres which solve the cauchy problem for their hessian equation as consequences we can characterize their geodesics and obtain a generalized symmetry principle then we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non flat affine metric moreover we present interesting examples with singular curves and isolated singularities

entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

trajectory sensitivity analysis using orthogonal functions using orthogonal functions it is shown that the problem of determining the trajectory sensitivity function reduces to that of solving a set of linear algebraic equations this approach is new and computationally simple

non removable singularities of a fourth order nonlinear partial differential equation abstract the aim of this paper is to give a classification of the solutions x y of the fourth order nonlinear equation y y x x 2 x y x y x x y y 0 det 2 3 4 in a punctured domain which are c 2 at the singularity we prove that this kind of solutions either have a removable singularity or are asymptotic to rotationally symmetric solutions near the singularity a geometric description of solutions which are not c 1 at the singularity is also given

orthogonal series approach to the sensitivity analysis of linear systems the orthogonal series expansion technique is applied for the approximate determination of sensitivity functions of linear dynamic systems as shown the present approach has two distinct advantages only a small number of coefficients need to be computed for the determination of all sensitivity functions and these coefficients are computed directly from simple equations the method is derived for single input linear systems and then extended to more general cases an illustrative example is included

ribaucour type transformations for the hessian one equation abstract we extend the classical theory of ribaucour transformations to the family of improper affine maps and use it to obtain new solutions of the hessian one equation we prove that such transformations produce complete embedded ends of parabolic type and curves of singularities which generically are cuspidal edges moreover we show that these ends and curves of singularities do no intersect we apply ribaucour transformations to some helicoidal improper affine maps providing new 3 parameter families with an interesting geometry and a good behavior at infinity in particular we construct improper affine maps periodic in one variable with any even number of complete embedded ends

sensitivity analysis of linear systems using orthogonal functions some useful properties of coefficients for determining the sensitivity functions of linear systems are given these properties simplify the numerical computations of sensitivity functions with respect to the system parameters

improper affine maps we study a class of improper affine spheres with singularities called improper affine maps new examples of genus 0 and 1 are described and the classification of complete improper affine maps with at most two embedded ends will be given

flat surfaces in the hyperbolic 3 space in this paper we give a conformal representation of flat surfaces in the hyperbolic 3 space using the complex structure induced by its second fundamental form we also study some examples and the behaviour at infinity of complete flat ends

a kinetic approach to comparison properties for degenerate parabolic hyperbolic equations with boundary conditions abstract we study the comparison principle for degenerate parabolic hyperbolic equations with initial and nonhomogeneous boundary conditions we prove a comparison theorem for any entropy sub and supersolution the l 1 contractivity and therefore uniqueness of entropy solutions has been obtained so far by some authors but it seems that any comparison theorem is not proven the method used there is the doubling variable technique due to kru kov our method is based upon the kinetic formulation and the kinetic techniques by developing the kinetic techniques for degenerate parabolic hyperbolic equations with boundary conditions we can obtain a comparison property which obviously extends the l 1 contractive property

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

the nonaccretive radiative transfer equations existence of solutions and rosseland approximation we present an existence theory and an asymptotic analysis for the radiative transfer equations ue t xuee ue e2 ue ue 0 inx 1 ue x sn k ue t 0 u0 where u u t x t r x x rn 1 sn and ue t x 1 sn ue t x d we prove that even if has a singularity 0 1 has a solution ue l r x sn as e 0 we show that ue converges pointwise to a function u l r x solution of the degenerate parabolic equation u t f u 0 in x u x k u t 0 u0 this is achieved without any monotonicity assumption on and therefore one cannot use the theory of nonlinear contraction semigroups

an extension of a theorem by k jorgens and a maximum principle at infinity for parabolic affine spheres where is a planar domain and f is in the usual holder space c2 without loss of generality we shall consider only locally convex solutions of 1 this equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere in short pa sphere in the unimodular affine real 3 space see c1 c2 cy and lsz contrary to the case of smooth bounded convex domains little is known about solutions of 1 when the domain is unbounded here we recall a famous result by k jorgens which asserts that any solution of 1 on r2 is a quadratic polynomial see j and we also mention a previous paper see fmm where the authors study solutions of 1 on the exterior of a planar domain that are regular at infinity since the underlying almost complex structure of 1 is integrable one expects pa spheres with their canonical conformal structure to be conveniently described in terms of meromorphic functions the reader will find in sect 2 a complex representation of pa spheres and particularly a complex description for the solutions of 1

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

the cauchy problem for improper affine spheres and the hessian one equation we give a conformal representation for improper affine spheres which is used to solve the cauchy problem for the hessian one equation with this representation we characterize the geodesics of an improper affine sphere study its symmetries and classify the helicoidal ones finally we obtain the complete classification of the isolated singularities of the hessian one monge ampere equation

a bernstein problem for special lagrangian equations in exterior domains abstract we establish quadratic asymptotics for solutions to special lagrangian equations with supercritical phases or with semiconvexity on solutions in exterior domains the method for convex case is based on an exterior evans krylov or exterior liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded hessian and also certain rotation arguments toward hessian bound our unified approach also leads to quadratic asymptotics for convex solutions to monge ampere equations previously known quadratic hessian equations and inverse harmonic hessian equations over exterior domains the semiconvex case is based on allard almgren s uniqueness of tangent cones

well posedness for stochastic scalar conservation laws with the initial boundary condition abstract in this paper we are interested in the initial non homogeneous dirichlet boundary value problem for a multi dimensional scalar non linear conservation law with a multiplicative stochastic forcing we introduce a notion of renormalized kinetic formulations in which the kinetic defect measures on the boundary of a domain are truncated in such a kinetic formulation we establish a result of well posedness of the initial boundary value problem under only the assumptions h 1 h 2 and h 3 stated below which are very similar ones in 6

stochastic scalar conservation laws we introduce a notion of stochastic entropic solution a la kruzkov but with ito s calculus replacing deterministic calculus this results in a rich family of stochastic inequalities defining what we mean by a solution a uniqueness theory is then developed following a stochastic generalization of l1 contraction estimate an existence theory is also developed by adapting compensated compactness arguments to stochastic setting we use approximating models of vanishing viscosity solution type for the construction while the uniqueness result applies to any spatial dimensions the existence result in the absence of special structural assumptions is restricted to one spatial dimension only

symmetry and uniqueness of parabolic affine spheres the proof of a variety of problems about symmetry properties in partial differential equations and differential geometry is based on the maximum principle for elliptic linear equations see a1 gnn and s in this paper we apply a similar technique to a locally strongly convex parabolic affine sphere in short pa sphere embedded in the unimodular affine real 3 space j 3 the study of pa spheres is locally equivalent see c1 c2 to the study of convex solutions of the monge amp re equation

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

the dirichlet problem for a conservation law with a multiplicative stochastic perturbation abstract in this paper we are interested in the dirichlet boundary value problem for a multi dimensional nonlinear conservation law with a multiplicative stochastic perturbation using the concept of measure valued solutions and kruzhkov s semi entropy formulations a result of existence and uniqueness of the entropy solution is proved

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

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

singularities of improper affine maps and their hessian equation abstract we study improper affine spheres with some admissible singularities called improper affine maps and associated to the unimodular hessian equation in particular we characterize when a curve of r 3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails also we consider improper affine maps with isolated singularities and show some similarities and differences between the hessian 1 equation and the hessian 1 equation as a consequence we construct global examples with the desired singularities

output sensitivity analysis using orthogonal functions a new method is proposed for the determination of the output sensitivity function this method is based on the idea of using orthogonal functions and has the distinct advantage that it reduces the problem of determining the output sensitivity function to that of solving linear algebraic equations the approach is straightforward and appears to be computationally simpler than known techniques

resolution d equations d evolution quasilineaires en dimension n d espace a l aide d equations lineaires en dimension n 1 on montre que l on peut calculer les solutions de l equation d evolution quasi lineaire dans r n d t u div b u a u 0 u x t r x r n t 0 et en particulier de l equation d ordre 1 d t u div b u 0 u x t r x r n t 0 a l aide d une equation d evolution lineaire dans r n r

a kinetic construction of global solutions of first order quasilinear equations on considere le probleme de cauchy pour des equations quasilineaires d ordre 1 de type conservation dans la region t 0 u t i 1 u a i u x i 0 u x o u o x ou x x 1 x n u u x t et a i i 1 n sont des fonctions continument differentiables a valeurs reelles d une seule variable reelle

double shifted chebyshev series for convolution integral and integral equations double shifted chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral volterra integral equation and fredholm integral equation this method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form in addition the computational complexity can be reduced remarkably three examples are illustrated it is seen that the proposed approach is straightforward and convenient and converges faster in finding approximations than other existing orthogonal function methods

scalar conservation laws with stochastic forcing we show that the cauchy problem for a randomly forced periodic multi dimensional scalar first order conservation law with additive or multiplicative noise is well posed it admits a unique solution characterized by a kinetic formulation of the problem which is the limit of the solution of the stochastic parabolic approximation

analysis and optimal control of time varying linear systems via shifted legendre polynomials this paper extends the application of shifted legendre polynomial expansion to time varying systems the extension is achieved through representing the product of two shifted legendre series in a new shifted legendre series with this treatment of the product of two time functions the operational properties of the shifted legendre polynomials are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index

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

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results