interaction of rarefaction waves of the two dimensional self similar euler equations we construct classical self similar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic euler equations in two space dimensions the binary interaction represents a major type of interaction in the two dimensional riemann problems and includes in particular the classical problem of the expansion of a wedge of gas into vacuum based on the hodograph transformation the method employed here involves the phase space analysis of a second order equation and the inversion back to or development onto the physical space

radon measure solutions for steady compressible euler equations of hypersonic limit conical flows and newton s sine squared law abstract we formulate a mathematical problem on hypersonic limit of three dimensional steady uniform non isentropic compressible euler flows of polytropic gases passing a straight cone with arbitrary cross section and attacking angle which is to study radon measure solutions of a nonlinear hyperbolic system of conservation laws on the unit 2 sphere the construction of a measure solution with density containing dirac measures supported on the surface of the cone is reduced to find a regular periodic solution of highly nonlinear and singular ordinary differential equations ode for a circular cone with zero attacking angle we then proved the newton s sine squared law by obtaining such a measure solution this provides a mathematical foundation for the newton s theory of pressure distribution on three dimensional bodies in hypersonic flows

the transonic shock in a nozzle 2 d and 3 d complete euler systems abstract in this paper we study a transonic shock problem for the euler flows through a class of 2 d or 3 d nozzles the nozzle is assumed to be symmetric in the diverging or converging part if the supersonic incoming flow is symmetric near the divergent or convergent part of the nozzle then as indicated in section 147 of r courant k o friedrichs supersonic flow and shock waves interscience publ new york 1948 there exist two constant pressures p 1 and p 2 with p 1 p 2 such that for given constant exit pressure p e p 1 p 2 a symmetric transonic shock exists uniquely in the nozzle and the position and the strength of the shock are completely determined by p e moreover it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting furthermore we establish the global existence stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure p e when the initial unsteady shock lies in the symmetric diverging part of the 2 d or 3 d nozzle on the other hand it is shown that an unsteady symmetric transonic shock is structurally unstable in a global in time sense if it lies in the symmetric converging part of the nozzle

on almost global existence for nonrelativistic wave equations in 3d almost global solutions are constructed to three dimensional quadratically nonlinear wave equations the proof relies on generalized energy estimates and a new decay estimate the method applies to equations that are only classically invariant such as the nonlinear system of hyperelasticity 1996 john wiley sons inc

variational principles for complex conductivity viscoelasticity and similar problems in media with complex moduli linear processes in media with dissipation arising in conductivity optics viscoelasticity etc are considered time periodic fields in such media are described by linear differential equations for complex valued potentials the properties of the media are characterized by complex valued tensors for example by complex conductivity or complex elasticity tensors variational formulations are suggested for such problems the functionals whose euler equations coincide with the original ones are constructed four equivalent variational principles are obtained two minimax and two minimal ones the functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation the last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors

on the global existence and stability of a multi dimensional supersonic conic shock wave we establish the global existence and stability of a three dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle the flow is described by a 3 d steady potential equation which is multi dimensional quasilinear and hyperbolic with respect to the supersonic direction making use of the geometric properties of the pointed shock surface together with the rankine hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new hardy type inequality on the shock surface based on this we prove that a multi dimensional conic shock attached to the vertex of the cone exists globally when the mach number of the incoming supersonic flow is sufficiently large moreover the asymptotic behavior of the 3 d supersonic conic shock solution which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone is also explicitly given

stability of the e h type regular shock refraction abstract in this paper the refraction of shocks on the interface for 2 d steady compressible flow is discussed particularly the e h type regular refraction is defined and its global stability of the wave structure is proved the 2 d steady potential flow equation is employed to describe the motion of the fluid the stability of the e h type regular refraction can be reduced to a free boundary problem of a nonlinear mixed type equation in an unbounded domain the corresponding linearized problem is a generalized tricomi problem of the lavrentiev bitsadze mixed type equation and it can be reduced to a nonlocal boundary value problem of an elliptic system the later is finally solved by establishing the bijection of a corresponding nonlocal operator in a weighted holder space the result obtained in this paper develops the theory of nonlinear mixed type equations and gives an application to a significant physical problem

non isentropic multi transonic solutions of euler poisson system abstract in this paper we consider the boundary value problem for the one dimensional non isentropic euler poisson system the physical subsonic inflow and outflow conditions are prescribed at boundaries we prove the existence of multi transonic solutions i e the existence of subsonic sonic supersonic shock subsonic flows in fact nozzles furthermore the monotonicity between the location of the transonic shock and the density at the exit of the nozzle will be elucidated

solution of the time dependent diffusion equation for layered diffusive media by the eigenfunction method an exact solution of the time dependent diffusion equation for the case of a two and a three layered finite diffusive medium is proposed the method is based on the decomposition of the fluence rate in a series of eigenfunctions and upon the solution of the consequent transcendental equation for the eigenvalues obtained from the boundary conditions comparisons among the solution of the diffusion equation and the results of monte carlo simulations show the correctness of the proposed model

exact solution of poisson s equation with an elliptical boundary abstract an exact green s function of the 2d poisson equation for an elliptical boundary is derived in terms of elementary functions which can be readily implemented and efficiently evaluated in addition the corresponding closed form solution is derived and used for studying the convergence the accuracy and the numerical stability of both expressions the green s function for the elliptic hole is also considered by using the joukowsky mapping in a modified form in addition the green s functions of the 2d poisson equation in the rectangular domain are presented in elementary functions

convection enhanced diffusion for random flows we analyze the effective diffusivity of a passive scalar in a two dimensional steady incompressible random flow that has mean zero and a stationary stream function we show that in the limit of small diffusivity or large peclet number with convection dominating there is substantial enhancement of the effective diffusivity our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory some of which are widely believed to be correct but have not been proved yet we show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of isichenko and kalda that the effective diffusivity behaves like 3 13 when the molecular diffusivity is small assuming some percolation theoretic facts we also analyze the effective diffusivity for a special class of convective flows random cellular flows where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows

global structure stability of riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary j math anal appl 330 1 2007 511 540 in the present paper we study the global structure stability of the riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the existence and uniqueness of global piecewise c 1 solution containing only shocks and contact discontinuities to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws in the half space t x t 0 x 0 our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in a half space possesses a global structural stability some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines are also given

global supersonic conic shock wave for the steady supersonic flow past a cone polytropic gas abstract in this paper we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value the flow is assumed to be polytropic isentropic and described by a steady potential equation based on the delicate asymptotic expansion of the background solution one can verify that the boundary conditions on the shock and the conic surface satisfy the dissipative property from this property by use of the reflected characteristics method and the special form of the shock equation we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large on the other hand we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution moreover our proof approach is different from that in shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132

transonic shock wave in an infinite nozzle asymptotically converging to a cylinder abstract we construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder which is close to a uniform transonic shock wave in other words suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily if we perturbed the supersonic incoming flow and the infinite nozzle a little bit we can obtain a transonic wave near the uniform one as a consequence we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall based on the theory of g q chen m feldman existence and stability of multi dimensional transonic flows through an infinite nozzle of arbitrary cross sections arch ration mech anal 184 2007 185 242 the crucial parts of this paper are to derive the uniform schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder

shape representation and classification using the poisson equation we present a novel approach that allows us to reliably compute many useful properties of a silhouette our approach assigns for every internal point of the silhouette a value reflecting the mean time required for a random walk beginning at the point to hit the boundaries this function can be computed by solving poisson s equation with the silhouette contours providing boundary conditions we show how this function can be used to reliably extract various shape properties including part structure and rough skeleton local orientation and aspect ratio of different parts and convex and concave sections of the boundaries in addition to this we discuss properties of the solution and show how to efficiently compute this solution using multigrid algorithms we demonstrate the utility of the extracted properties by using them for shape classification and retrieval

global smooth solutions of 3 d null form wave equations in exterior domains with neumann boundary conditions abstract the paper is devoted to investigating long time behavior of smooth small data solutions to 3 d quasilinear wave equations outside of compact convex obstacles with neumann boundary conditions concretely speaking when the surface of a 3 d compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition we prove that the smooth small data solution exists globally provided that the neumann boundary condition on the exterior domain is given one of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself as an application of the main result the global stability to 3 d static compressible chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude

the closed form solution for the 2d poisson equation with a rectangular boundary the 2d green s function for the poisson equation with a rectangular boundary is investigated using the schwarz christoffel transformation method a closed form of the green s function containing jacobi elliptic functions is developed

on the instability problem of a 3 d transonic oblique shock wave abstract in this paper we are concerned with the instability problem of a 3 d transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge the flow is assumed to be isentropic and irrotational it was indicated on p 317 of 7 that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge then it follows from the rankine hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge which corresponds to a supersonic shock or a transonic shock respectively the question arises which of the two actually occurs it has frequently been stated that the strong one is unstable and that therefore only the weak one could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand such a longstanding open question we will show that the attached 3 d transonic oblique shock problem is overdetermined with respect to the periodic perturbation which implies that the 3 d transonic shock is unstable in general

global shock waves for the supersonic flow past a perturbed cone we prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone the flow is governed by the potential equation as well as the boundary conditions on the shock and the surface of the body it is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self similar solution under some suitable conditions our analysis is based on a global uniform weighted energy estimate for the linearized problem combining this with the local existence result of chen li 1 we establish the global existence and decay rate of the solution to the nonlinear problem

nonlinear stability of a self similar 3 dimensional gas flow we show that the 3 dimensional supersonic gas flow past an infinite cone is nonlinear staple upon the perturbation of the obstacle the perturbed flow exists globally in space and tends to the self similar flow downstream there is a thin layer of concentration of vorticities and entropy variation our analysis is based on an approximation scheme using local self similar solutions as building blocks this enables us to obtain global estimates of the nonlinear interactions of waves needed for the stability analysis

sonic supersonic solutions for the two dimensional steady full euler equations this paper focuses on the structure of classical sonic supersonic solutions near sonic curves for the two dimensional full euler equations in gas dynamics in order to deal with the parabolic degeneracy near the sonic curve a novel set of dependent and independent variables are introduced to transform the euler equations into a new system of governing equations which displays a clear regularity singularity structure with the help of technical characteristic decompositions the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables this is the first construction of a classical sonic supersonic solution near a sonic curve for the full euler equations

network approximation for transport properties of high contrast materials we show that the effective complex impedance of materials with conductivity and dielectric permittivity that have high contrast can be calculated approximately by solving a suitable resistor capacitor network we use variational principles extensively for the analysis and we assess the accuracy of the network approximation by numerical computations

global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone in this paper we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body the flow is assumed to be polytropic isentropic and described by a steady potential equation theoretically as indicated in r courant k o friedrichs supersonic flow and shock waves interscience publishers inc new york 1948 it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface which correspond to the supersonic shock and the transonic shock respectively in the references shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone polytropic case preprint 2006 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone isothermal case pacific j math 233 2 2007 257 289 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132 the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow at present for the supersonic symmetric incoming flow we will study the global transonic shock problem when the pressure at infinity is appropriately large

steady supersonic flow past a curved cone abstract this paper studies the steady supersonic flow past a lipschitz curved cone under the assumptions that the cone has an opening angle less than a critical value and has sufficiently small total variation of the tangent of the perturbation and that the mach number of incoming flow is sufficiently large the global weak solution is constructed via glimm scheme for 1 3

on a degenerate mixed type boundary value problem to the 2 d steady euler equation abstract this paper is concerned with the structure of sonic supersonic solutions near sonic curves arising from the transonic channel flow problems in gas dynamics given two smooth curves one of which is a sonic curve and the other is a characteristic curve we construct a local classical supersonic solution for the two dimensional steady euler equations in the angular region a novel set of dependent and independent variables are introduced to transform the euler equations into a linear system with explicitly singularity regularity structures we use the iteration method to establish the existence and uniqueness of smooth solutions for the new system and then express the solution in terms of the original variables

global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary abstract this work is a continuation of our previous work in the present paper we study the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws with non linear boundary conditions in the half space t x t 0 x 0 under the assumption that each characteristic with positive velocity is linearly degenerate we prove the existence and uniqueness of global weakly discontinuous solution u u t x with small amplitude and this solution possesses a global structure similar to that of the self similar solution u u x t of the corresponding riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines particularly to the system describing the motion of the relativistic string in minkowski space r 1 n are also given

existence and stability of supersonic euler flows past lipschitz wedges it is well known that when the vertex angle of a straight wedge is less than the critical angle there exists a shock front emanating from the wedge vertex so that the constant states on both sides of the shock front are supersonic since the shock front at the vertex is usually strong especially when the vertex angle of the wedge is large then a global flow is physically required to be governed by the isentropic or adiabatic euler equations in this paper we systematically study two dimensional steady supersonic euler i e nonpotential flows past lipschitz wedges and establish the existence and stability of supersonic euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small we develop a modified glimm difference scheme and identify a glimm type functional by naturally incorporating the lipschitz wedge boundary and the strong shock front and by tracing the interaction not only between the boundary and weak waves but also between the strong shock front and weak waves to obtain the required bv estimates these estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock fronts emanating from the vertex to the strong shock front of the entropy solution the regularity of strong shock fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established