Formation of singularities for viscosity solutions of hamilton—jacobi equations in one space variable

In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in one space variable, under no assumption on the convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which is the correct class of weak solutions. We obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing explicitly the proper rarefaction waves.     

Well-posedness of the cauchy problem for the coupled system of the Schrödinger-KdV equations

This paper is devoted to the study of the Cauchy problem for the coupled system of the Schrödinger-KdV equations which describes the nonlinear dynamics of the one-dimensional Langmuir and ion-acoustic waves. Global well-posedness of the problem is established in the spaceH ^κ ×H ^κ (κ εZ ⁺), the first and second components of which correspond to the electric field of the Langmuir oscillations and the low-frequency density perturbation respectively.     

Optimal control of diffusion processes and hamilton–jacobi–bellman equations part 2 : viscosity solutions and uniqueness

We consider general optimal stochastic control problems and the associated Hamilton–Jacobi–Bellman equations. We develop a general notion of week solutions – called viscosity solutions – of the amilton–Jocobi–Bellman equations that is stable and we show that the optimal cost functions of the control problems are always solutions in that sense of the Hamilton–Jacobi–Bellman equations. We then prove general uniqueness results for viscosity solutions of the Hamilton–Jacobi–Bellman equations.     

Traveling wave solutions for Generalized Drinfeld–Sokolov equations

In this paper, solitary waves and periodic waves for Generalized Drinfeld–Sokolov equations are studied, by using the theory of dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained.     

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.     

On the cauchy problem for some non-Kowalewskian equations inD {σ} L2 (*)

A pseudo-differential operator is considered, which generalizes some peculiar non-Kowalewskian operators of 2-evolution type. A result is proved about the well-posedness of the Cauchy problem inD _{} ^L2 , where 1 is Gevrey index.     

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ^−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.     

Local existence of energy class solutions for the dirac—klein—gordon equations

Abstract

We show a method to eliminate a type of mixed asymptotics in certain free boundary problems, and give two examples of its application. It appears that these problems cannot be handled by the monotonicity formula of Alt et al. [Alt, H. W., Caffarelli, L. A., Friedman, A. (1984). Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2):431–461] or by the more recent monotonicity formula of Caffarelli et al. [Caffarelli, L. A., Jerison, D., Kenig, C. E. (2002). Some new monotonicity theorems with applications to free boundary problems. Ann. Math. (2) 155(2):369–404].     

On uniqueness of solutions to the Cauchy problem for degenerate Fokker–Planck–Kolmogorov equations

We prove a new uniqueness result for highly degenerate second-order parabolic equations on the whole space. A novelty is also our class of solutions in which uniqueness holds.     

New soliton solutions of the generalized Zakharov equations using He’s variational approach

In this paper, we obtain new soliton solutions of the generalized Zakharov equations by the well-known He’s variational approach. The condition for continuation of the new solitary solution is obtained.     

Criterion for the solvability of the cauchy problem for an abstract Euler–Poisson–Darboux equation

We consider the Cauchy problem for an abstract Euler–Poisson–Darboux equation in a Banach space and prove a necessary and a sufficient condition for the solvability of this problem. The conditions are stated in terms of an estimate for the norm of a fractional power of the resolvent and its derivatives. The properties of solutions are established, and examples are given.     

Generalized Convexity and Generalized Quasi—variational Inequality

51.PreliminaryLetXbeaset.Througho11ttl1lspaperweshalldenoteby2xthefaruilyofallsubsetsofX,by(X)thefamilyofallnoneruptyfinitesuhsetsof'X.lfXlsasul,setofavectorspace,co(X)denotestheconvexhu1lofX.IfXisatopologicalspaceandA=X,thenwedenotebyintx(A)theil1teriorofAinXandc.l.(A)theclosureofAinXrespectively.Definitionl.1LetXandYl)etoI3ologicalspacesandT:X-2"isaset-vail1edmapping.(i)Tissaidtobeuppersemicontin\1ous(resp.lowersemicontinuo1Js)atxo6X,ifforeachopensetGinYwithG=T(x,)(resI3.Gn7'(x,…     

Semicontinuous solutions for Hamilton-Jacobi equations and theL ∞control problem

We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL -control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form –u _t +H(t, x, u, –Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.Supported in part by Grant DMS-9300805 from the National Science Foundation.     

Nonexistence of global solutions of the Cauchy problem for systems of klein–Gordon equations with positive initial energy

We study the Cauchy problem for systems of weakly coupled Klein–Gordon equations with dissipations. We prove a theorem on the nonexistence of global solutions with positive initial energy.     

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

For the 5-components Maxwell–Bloch system the stability problem for the isolated equilibria is completely solved. Using the geometry of the symplectic leaves, a detailed construction of the homoclinic orbits is given. Studying the problem of invariant sets for the system, we discover a rich family of periodic solutions in explicit form.