rotational degeneracy of hyperbolic systems of conservation laws the purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy in this situation hyperbolicity is necessarily non strict so that the elementary waves have interesting patterns the discussion is centered around a theorem on existence and uniqueness of solutions of the riemann problem we give a unified presentation of examples from continuum mechanics

transonic shocks in multidimensional divergent nozzles we establish existence uniqueness and stability of transonic shocks for a steady compressible non isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross section for a prescribed exit pressure the proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation in the process we obtain unique solvability for a class of transport equations with velocity fields of weak regularity non lipschitz an infinite dimensional weak implicit mapping theorem which does not require continuous frechet differentiability and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions

order and stepsize control in extrapolation methods the paper presents a new theory for joint order and stepsize control in extrapolation methods this theory defines a locally optimal order that can be determined along any trajectory to be computed in addition shannon s information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non stiff integration by the new device moreover a significant increase in reliability robustness and portability of the extrapolation codes is achieved

stability of a mach configuration we prove the stability of a mach configuration which occurs in shock reflection off an obstacle or shock interaction in compressible flow the compressible flow is described by a full steady euler system of gas dynamics the unperturbed mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity among four regions divided by these four lines in the neighborhood of the intersection two are supersonic regions and other two are subsonic regions we prove that if the constant states in the supersonic regions are slightly perturbed then the structure of the whole configuration holds while the other two shock fronts and the slip line as well as the flow field in the subsonic regions are also slightly perturbed such a conclusion asserts the existence and stability of the general mach configuration in shock theory in order to prove the result we reduce the problem to a free boundary value problem where two unknown shock fronts are free boundaries while the slip line is transformed to a fixed line by a lagrange transformation in the region where the solution is to be determined we have to deal with an elliptic hyperbolic composed system by decoupling this system and combining the technique for both hyperbolic equations and elliptic equations we establish the required estimates which are crucial in the proof of the existence of a solution to the free boundary value problem 2005 wiley periodicals inc

quasi one dimensional riemann problems and their role in self similar two dimensional problems we study two dimensional riemann problems with piecewise constant data we identify a class of two dimensional systems including many standard equations of compressible flow which are simplified by a transformation to similarity variables for equations in this class a two dimensional riemann problem with sectorially constant data becomes a boundary value problem in the finite plane for data leading to shock interactions this problem separates into two parts a quasi one dimensional problem in supersonic regions and an equation of mixed type in subsonic regions we prove a theorem on local existence of solutions of quasi one dimensional riemann problems for 2 2 systems we generalize a theorem of courant friedrichs that any hyperbolic state adjacent to a constant state must be a simple wave in the subsonic regions where the governing equation is of mixed hyperbolic elliptic type we show that the elliptic part is degenerate at the boundary with a nonlinear variant of a degeneracy first described by keldysh

convergence of the fractional step lax friedrichs scheme and godunov scheme for the isentropic system of gas dynamics a convergence theorem of the fractional step lax friedrichs scheme and godunov scheme for an inhomogeneous system of isentropic gas dynamics 1 5 3 is established by using the framework of compensated compactness meanwhile a corresponding existence theorem of global solutions with large data containing the vacuum is obtained

stability of strong travelling waves for a non local time delayed reaction diffusion equation the paper is concerned with a non local time delayed reaction diffusion equation we prove the time asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength i e the so called strong wavefront by means of the technical weighted energy method when the initial perturbation around the wave is small the exponential convergent rate is also given selection of a suitable weight plays a key role in the proof

rotationally invariant hyperbolic waves we use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves the equation can include weak dissipative dispersive or diffractive effects we give applications to equations from magnetohydrodynamics elasticity and viscoelasticity

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

global uniqueness of transonic shocks in two dimensional steady compressible euler flows we prove that for the two dimensional steady complete compressible euler system with given uniform upcoming supersonic flows the following three fundamental flow patterns special solutions in gas dynamics involving transonic shocks are all unique in the class of piecewise c1 smooth functions under appropriate conditions on the downstream subsonic flows i the normal transonic shocks in a straight duct with finite or infinite length after fixing a point the shock front passing through ii the oblique transonic shocks attached to an infinite wedge iii a flat mach configuration containing one supersonic shock two transonic shocks and a contact discontinuity after fixing a point where the four discontinuities intersect these special solutions are constructed traditionally under the assumption that they are piecewise constant and they have played important roles in the studies of mathematical gas dynamics our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows which are sufficient to uniquely determine these special solutions mathematically these are uniqueness results on solutions of free boundary problems of a quasi linear system of elliptic hyperbolic composite mixed type in bounded or unbounded planar domains without any assumptions on smallness the proof relies on an elliptic system of pressure p and the tangent of the flow angle w v u obtained by decomposition of the euler system in lagrangian coordinates and a newly developed method for the l estimate that is independent of the free boundaries by combining the maximum principles of elliptic equations and careful analysis of the shock polar applied on the maybe curved shock fronts

stability of supersonic contact discontinuity for two dimensional steady compressible euler flows in a finite nozzle abstract in this paper we study the stability of supersonic contact discontinuity for the two dimensional steady compressible euler flows in a finitely long nozzle of varying cross sections we formulate the problem as an initial boundary value problem with the contact discontinuity as a free boundary to deal with the free boundary value problem we employ the lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem we develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain finally by using the inverse lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity

existence and uniqueness analysis of a detached shock problem for the potential flow abstract we study a problem for two dimensional steady potential and isentropic euler equations in a bounded domain where an artificial detached shock interacts with a wedge using the stream function we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted holder spaces the proof is based on various holder and schauder estimates for second order elliptic equations and fixed point theorems moreover we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional

transonic shock in finitely long nozzle with porous medium boundary condition in this paper we investigate the 2 d euler equations with complex boundary conditions for the entrance and exit of finitely long nozzle we use the supersonic incoming flow condition and the end pressure condition respectively in addition for the nozzle wall the lower one is solid and the upper one is permitted to have leakage we establish the well posedness if appropriate seepage discharge is given we also obtain the shock position by solving a free boundary problem without assuming the shock must go through a fixed point

pulsating type entire solutions of monostable reaction advection diffusion equations in periodic excitable media abstract we establish the existence of pulsating type entire solutions of reaction advection diffusion equations with monostable nonlinearities in a periodic framework here the nonlinearities include the classic kpp case the pulsating type entire solutions are defined in the whole space and for all time t r by studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions and approaching each other the key techniques are to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions

on subsonic compressible flows in a two dimensional duct abstract this paper concerns subsonic flows passing a two dimensional duct for the steady compressible euler system if the bernoulli constant is uniform in the flow field the density at the entry and both the pressures at the entrance and the exit are given we show that the problem is generally ill posed but if we give the pressure at the exit with a constant difference then under the same other conditions as above we establish the existence of subsonic flows

stability of fronts for a kpp system ii the critical case abstract we study a system of n 1 coupled semilinear parabolic equations on the real line which depends on a small parameter and reduces to the scalar kolmogorov petrovsky piscounov kpp equation when 0 under appropriate scaling the system has a family of traveling fronts parametrized by their speed when 2 as in the scalar kpp case the case of critical speed 2 say is investigated and it is shown that the system inherits some crucial properties of the kpp equation when is small in particular the asymptotic stability of the front in a local and semiglobal sense first we describe the properties of the front and then apply functional arguments to prove its local stability in an adequate weighted sobolev space moreover the decay rate of the perturbations is shown to be polynomial in time finally we show a semiglobal stability property of the front which also is inherited from the scalar kpp equation

speed up of combustion fronts in shear flows this paper is concerned with the analysis of speed up of reaction diffusion advection traveling fronts in infinite cylinders with periodic boundary conditions the advection is a shear flow with a large amplitude and the reaction is nonnegative with either positive or zero ignition temperature the unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity solving an open problem from berestycki nonlinear pdes in condensed matter and reactive flows kluwer doordrecht 2003 the asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional hormander type condition on the flow

entire solutions of reaction advection diffusion equations with bistable nonlinearity in cylinders this paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction advection diffusion equation with infinite cylinders assume that the equation admits three equilibria two stable equilibria 0 and 1 and an unstable equilibrium it is well known that there are different wave fronts connecting any two of those three equilibria by considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions we establish three different types of entire solutions finally we analyze a model for shear flows in cylinders to illustrate our main results

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

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

local and nonlocal discrete regularization on weighted graphs for image and mesh processing we propose a discrete regularization framework on weighted graphs of arbitrary topology which unifies local and nonlocal processing of images meshes and more generally discrete data the approach considers the problem as a variational one which consists in minimizing a weighted sum of two energy terms a regularization one that uses the discrete p dirichlet form and an approximation one the proposed model is parametrized by the degree p of regularity by the graph structure and by the weight function the minimization solution leads to a family of simple linear and nonlinear processing methods in particular this family includes the exact expression or the discrete version of several neighborhood filters such as the bilateral and the nonlocal means filter in the context of images local and nonlocal regularizations based on the total variation models are the continuous analog of the proposed model indirectly and naturally it provides a discrete extension of these regularization methods for any discrete data or functions

partitioning ode systems with an application to air pollution models abstract the numerical treatment of stiff ode systems is carried out by using implicit methods a long sequence of nonlinear systems has to be treated when implicit methods are used the newton iterative method is often used in the solution of these systems this leads to the calculation of jacobian matrices and to the inversion of these matrices the computational work can in some cases be reduced considerably when some kind of partitioning is used the conditions under which the partitioning procedures can successfully be used will be studied in this paper an example taken from a large air pollution model will be given to illustrate the usefulness of the theoretical results

singular limit and homogenization for flame propagation in periodic excitable media this paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media the model involves two parameters the period of the medium l and a singular parameter related to the activation energy the existence of pulsating travelling fronts for fixed and l was proved by berestycki hamel in bh in the present paper we investigate the behaviour of such solutions when more precisely we establish that pulsating travelling fronts behave like travelling waves when the period l is small and we also study the convergence of the solution as goes to zero and l is fixed toward a solution of a free boundary problem

stability of travelling fronts in a model for flame propagation part ii nonlinear stability this paper is concerned with the nonlinear stability of the travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number the model consists in a semilinear parabolic equation in an infinite cylinder with neumann boundary conditions we prove that any solution which is initially close to a travelling wave will converge to a translate of that wave

relationships between emission sources and excess ozone concentrations while the ozone layer in the stratosphere is shielding the earth s surface from radioactivity high ozone concentrations in the troposphere have damaging effects on plants animals and humans when they exceed certain critical levels this is why it is necessary to resolve successfully three tasks 1 i to determine in a sufficiently accurate way the critical levels of the ozone concentrations 2 ii to reduce the ozone concentrations to the critical levels and 3 iii to develop reliable and robust control strategies by which the ozone concentrations can be kept under the critical levels both the second task and the third task are very expensive therefore the high ozone concentrations must be reduced to the critical levels but no more furthermore it is necessary to reduce as much as possible the economical costs of these actions mathematical models can be used in the search of an optimal solution of the tasks listed above a mathematical model for studying air pollution levels in europe the danish eulerian model will be shortly discussed the model consists of a system of 35 partial differential equations and leads to huge computational tasks when it is handled numerically therefore it is necessary to use big modern high speed computers with vector and or parallel capabilities the danish eulerian model has been used to run various scenarios in order to study some of the relationships between emission sources in europe and high ozone concentrations modern visualization techniques have been applied to illustrate better the effects of different emission reductions used in the scenarios on the ozone concentrations and to draw some conclusions

stability of travelling fronts in a model for flame propagation part i linear analysis we investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit lewis number this model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder with neumann boundary conditions in this paper we prove that every travelling wave solution of this model is linearly stable our tools are exponential decay estimates for solutions of elliptic equations in a cylinder and the maximum principle for parabolic equations

solutions a variations bornees pour certains systemes hyperboliques de lois de conservation abstract blake temple trans amer math soc 280 1983 781 795 has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide in this note we prove the global existence of weak solutions for such systems with any bounded variation initial condition the proof is based upon standard numerical schemes as well as upon parabolic regularization the key is that the total variation of the riemann invariants is decreasing in time at least in the case of the initial condition with compact support we prove by using the glimm scheme that the system is decoupling in two conservation laws in one unknown in finite time

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

on the linearized stability of viscous shock profiles for systems of conservation laws abstract this paper is concerned with the linearized stability of traveling wave solutions for systems of viscous hyperbolic conservation laws the main purpose is to show that for a given traveling wave with shock profile from any characteristic family there exists an appropriate weighted norm space such that the traveling wave is exponentially stable in this space as a consequence if the initial disturbance has average zero and decays exponentially fast as x then the corresponding solution of the linearized equation decays to zero exponentially fast in t on any compact interval in x the proof is given by applying an elementary weighted characteristic energy method to the integrated linearized system based on the underlying wave structure

transonic nozzle flows and free boundary problems for the full euler equations abstract we establish the existence and uniqueness of transonic flows with a transonic shock through a two dimensional nozzle of slowly varying cross sections the transonic flow is governed by the steady full euler equations given an incoming smooth flow that is close to a constant supersonic state i e smooth cauchy data at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow this problem is approached by a one phase free boundary problem in which the transonic shock is formulated as a free boundary the full euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem an iteration scheme is developed and its fixed point is shown to exist which is a solution of the free boundary problem by combining some delicate estimates for the elliptic equation and the system of transport equations with the schauder fixed point argument the uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of euler lagrange type and detailed estimates of the solutions

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

one step and extrapolation methods for differential algebraic systems the paper analyzes one step methods for differential algebraic equations dae in terms of convergence order in view of extrapolation methods certain perturbed asymptotic expansions are shown to hold for the special dae extrapolation solver based on the semi implicit euler discretization the perturbed order pattern of the extrapolation tableau is derived in detail the theoretical results lead to modifications of the known code the efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion

riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation which is used to model the transition from regular to mach reflection for weak shock waves the related initial value problem consists of oblique shock data in the upper half plane with two parameters a and b corresponding to the slopes of the initial shock waves the study of quasi steady solutions leads to a problem that changes type when written in self similar coordinates the problem is hyperbolic in the region where the flow is supersonic and elliptic where the flow is subsonic in this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi one dimensional riemann problems in the region of physical space where the flow is subsonic we pose the related free boundary problems and discuss the behavior of the subsonic solution using results from our previo

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

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

transonic shocks for steady euler flows with cylindrical symmetry abstract in this paper we prove the existence of transonic shocks adjacent to a uniform one for the full euler system for steady compressible fluids with cylindrical symmetry in a cylinder and consequently show the stability of such uniform transonic shocks mathematically we solve a free boundary problem for a quasi linear elliptic hyperbolic composite system this reveals that the boundary conditions and equations interact in a subtle way the key point is to separate in a suitable way the elliptic and hyperbolic parts of the system the approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full euler system

on transonic shocks in a conic divergent nozzle with axi symmetric exit pressures abstract in this paper we establish the existence and stability of a 3 d transonic shock solution to the full steady compressible euler system in a class of de laval nozzles with a conic divergent part when a given variable axi symmetric exit pressure lies in a suitable scope thus for this class of nozzles we have solved such a transonic shock problem in the axi symmetric case described by courant and friedrichs 1948 in section 147 of 8 given the appropriately large exit pressure p e x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e x

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

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

existence of steady subsonic euler flows through infinitely long periodic nozzles abstract in this paper we study the global existence of steady subsonic euler flows through infinitely long nozzles which are periodic in x 1 direction with the period l it is shown that when the variation of bernoulli function at some given section is small and mass flux is in a suitable regime there exists a unique global subsonic flow in the nozzle furthermore the flow is also periodic in x 1 direction with the period l if in particular the bernoulli function is a constant we also get the existence of subsonic sonic flows when the mass flux takes the critical value

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

spatial decay and stability of traveling fronts for degenerate fisher type equations in cylinder abstract this paper is concerned with the spatial decay and asymptotic stability of multidimensional traveling fronts for the degenerate fisher type equation in an unbounded cylinder firstly by applying the moving plane argument and the generalized center manifold theorem we obtain the uniqueness and exponential decay of the traveling front with the critical speed and the non exponential decay of traveling fronts with non critical speeds especially for the p degree fisher equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non critical speeds secondly by applying the spectral analysis and sub super solution method we prove the nonlinear exponential stability of all traveling fronts in some exponentially weighted spaces and the lyapunov stability of traveling fronts with non critical speeds in some polynomially weighted spaces finally by combining the sub super solution method with the nonlinear stability results we get the asymptotic behavior and the asymptotic spreading speeds of the solution for more general initial data which are proved to be determined by the spatial decay of the initial data at one end

mach configuration in pseudo stationary compressible flow when a shock front in unsteady compressible flow is moving along an inclined ramp the incident shock will be reflected by the ramp as it moves forward in the process of reflection various reflected wave pattern may occur depending on the slope of the ramp and the strength of the incident shock 14 23 this problem is one of the most important multidimensional prototype problems in gas dynamics generally if the angle of inclination is larger than a critical value then the reflected shock starting from the intersection of the incident shock with the ramp forms a smooth bubble which expands when time goes on such a reflection is called regular reflection on the other hand if the angle of inclination is small then the intersection of the incident shock and the reflected shock will not touch the ramp and an additional shock front called mach stem appears which connects the ramp with the intersection of the incident shock and the reflected shock in this case three shock fronts meet at one point and form a triple shock configuration an important fact is that the structure solely containing three shock fronts separating the neighborhood of the triple intersection into three zones with different continuous states does not exist 14 27 many physical experiments indicate that there is a slip line issuing from the triple intersection such a local wave pattern is called mach configuration or mach structure correspondingly the whole process of reflection is called mach reflection because of the complexity of the whole picture of mach reflection it is natural and necessary to start work on getting a clear understanding of the local mach configuration mach reflection also occurs in various other cases for compressible flow for instance in the study on oblique shock reflection in stationary compressible flow von neumann indicated that see 14 25 if the incident shock hit a wall with an angle larger than a critical value then the reflection must be mach reflection many physical experiments approximate analysis and numerical computations verified the appearance of mach configuration and a tremendous amount of literature is devoted to study such problems see 3 4 11 19 20 24 and the references therein however for the rigorous proof on the stability of mach configuration we can only mention a related result on stationary flow in 13

transonic shocks for the full compressible euler system in a general two dimensional de laval nozzle in this paper we study the transonic shock problem for the full compressible euler system in a general two dimensional de laval nozzle as proposed in courant and friedrichs supersonic flow and shock waves interscience new york 1948 given the appropriately large exit pressure pe x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe x we solve this problem completely for a general class of de laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible euler system the problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic elliptic system one of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted holder space with low regularities for a second order quasilinear elliptic equation with a free parameter the position of the shock curve at one wall of the nozzle and non local terms involving the trace on the shock of the first order derivatives of the unknown function

existence of stationary supersonic flows past a pointed body in this paper we study the mathematical aspects of the stationary supersonic flow past a non axisymmetric curved pointed body the flow is described by a steady potential flow equation which is a quasilinear hyperbolic equation of second order we prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body provided the pointed body is a perturbation of a circular cone and the vertex angle of the approximate cone of the pointed body is less than a critical value the solution is smooth in between the shock and the surface of the body consequently such a structure of flow near the tip of the pointed body and its stability is verified mathematically

three dimensional steady subsonic euler flows in bounded nozzles abstract the existence and uniqueness of three dimensional steady subsonic euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit if in addition the normal component of the voriticity and the variation of bernoulli s function at the entrance are both zero then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number as the magnitude of the normal component of the momentum approaches the critical number the associated flows converge to a subsonic sonic flow furthermore when the normal component of vorticity and the variation of bernoulli function are both small the existence and uniqueness of subsonic euler flows with non zero vorticity are established the proof of these results is based on a new formulation for the euler system a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions detailed study for a linear div curl system and delicate estimate for the transport equations

two dimensional relativistic euler equations in a convex duct abstract in this paper we investigate the motion of supersonic flow governed by 2 d steady isentropic irrotational relativistic euler system entered into a convex duct we will give the propagation and interactions of rarefaction waves between the upward and downward wall and obtain the global existence of piecewise continuous solutions furthermore in some cases the vacuum state may appear and we also give a sufficient condition to the appearance of the vacuum in finite domain

existence of pulsating waves of advection reaction diffusion equations of ignition type by a new method abstract the goal of this paper is to study the advection reaction diffusion equation of ignition type u t u q x y u f u xin j x xin existence of planar flame fronts in convective diffusive periodic media arch ration mech anal 121 1992 205 233 berestycki and hamel h berestycki f hamel front propagation in periodic excitable media comm pure appl math 55 2002 949 1032 proved the existence of pulsating waves using the theory of degenerate elliptic equations our goal is to give an alternative and rather natural proof of the same result first we transform the problem so that the existence of pulsating waves is equivalent to the existence of 1 periodic in time solutions of a nonlinear parabolic equation with 1 periodic in time coefficients next we prove the existence of such periodic solutions using a continuation method based on the implicit function theorem

stability of transonic shock fronts in two dimensional euler systems we study the stability of stationary transonic shock fronts under two dimensional perturbation in gas dynamics the motion of the gas is described by the full euler system the system is hyperbolic ahead of the shock front and is a hyperbolic elliptic composed system behind the shock front the stability of the shock front and the downstream flow under two dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic elliptic composed system we develop a method to deal with boundary value problems for such systems the crucial point is to decompose the system to a canonical form in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients by several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem our result indicates the stability of the transonic shock front and the flow field behind the shock

steady euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles abstract we establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets entropy waves for the steady euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles we first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity or the bernoulli and entropy functions at the inlet for the smooth case then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets entropy waves by nonlinear arguments this is the first result on the global existence of solutions of the multidimensional steady compressible full euler equations with free boundaries which are not necessarily small perturbations of piecewise constant background solutions the subsonic sonic limit of the solutions is also shown finally through the incompressible limit we establish the existence and uniqueness of incompressible euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets the methods and techniques developed here will be useful for solving other problems involving similar difficulties

improved quasi steady state approximation methods for atmospheric chemistry integration in the last fifteen years the quasi steady state approximation qssa method has been a commonly used method for integrating stiff ordinary differential equations arising from atmospheric chemistry problems in this paper a theoretical analysis of the qssa method is developed stressing its strengths and its weaknesses this theory leads to practical improvements to the qssa method new algorithms including symmetric and extrapolated qssa are presented

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

stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations abstract we develop a method that works in general product riemannian manifold to decompose the three dimensional steady full compressible euler system which is of elliptic hyperbolic composite mixed type for subsonic flows the method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space r 3 under multidimensional perturbations of boundary conditions

transonic shocks in 3 d compressible flow passing a duct with a general section for euler systems this paper is devoted to the study of a transonic shock in three dimensional steady compressible flow passing a duct with a general section the flow is described by the steady full euler system which is purely hyperbolic in the supersonic region and is of elliptic hyperbolic type in the subsonic region the upstream flow at the entrance of the duct is a uniform supersonic one adding a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic hyperbolic system the new ingredients of our paper contain the decomposition of the elliptic hyperbolic system the determination of the shock front by a pair of partial differential equations coupled with the three dimensional euler system and the regularity analysis of solutions to the boundary value problems introduced in our discussion

entire solutions in monostable reaction advection diffusion equations in infinite cylinders abstract this paper is concerned with entire solutions of a monostable reaction advection diffusion equation in infinite cylinders without the condition f u f 0 by constructing a quasi invariant manifold we prove that there exist two classes of entire solutions furthermore we show that one class of such entire solutions is unique up to space and time translation

traveling waves for a boundary reaction diffusion equation abstract we prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature a key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction diffusion equation the so called high energy activation energy limit this explicit formula which is interesting in itself also allows us to get an estimate on the decay at infinity of the traveling wave which turns out to be faster than the usual exponential decay

existence and asymptotics of travelling waves in a thermo diffusive model in half cylinders part ii oblique boundary conditions abstract in this paper we investigate a system of reaction diffusion equations set in half cylinders and involving oblique boundary conditions this type of system arises in the modelization of burner flames and approach in a suitable sense conical shaped fronts in the whole space we prove the existence of solutions and some properties of the speed of propagation of the travelling fronts in some asymptotic limit

existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross sections we establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross sections including a slowly varying de laval nozzle the transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance uniform subsonic downstream flow at the exit at infinity and the slip boundary condition on the nozzle boundary our results indicate that if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow there exists a solution that consists of a c1 subsonic flow in the unbounded downstream region converging to a uniform velocity state at infinity and a c1 multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow and the shock is orthogonal to the nozzle boundary at every point of their intersection in order to construct such a transonic flow we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase in which the equation is elliptic and the free boundary is a transonic shock the free boundary conditions are determined by the rankine hugoniot conditions along the shock we further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain we also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance

compressible flow and transonic shock in a diverging nozzle the paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible euler system consider a nozzle having a shape as a diverging truncated sector with generic opening angle if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned then a transonic shock appears in the nozzle to determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation we prove that the receiver pressure can uniquely determine the location of the transonic shock as well as the flow behind the shock such a conclusion was conjectured by courant and friedrichs and is confirmed theoretically in this paper for the divergent nozzles

stability of reflection and refraction of shocks on interface abstract in this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media the attack will produce a reflected wave and a refracted wave and also let the interface deflected in this paper we will mainly study the case when the reflected wave is a shock and the flow between the reflected wave and the refracted shock is relatively subsonic our result indicates that the wave structure and the flow field for the reflection refraction problem in this case is conditionally stable to describe the motion of the fluid we use the inviscid euler system as the mathematical model the reflection refraction problem can be reduced to a free boundary value problem where the unknown reflected shock and refracted shock are free boundaries and the deflected interface is also to be determined in the proof of the existence and the stability of the corresponding wave structure we apply the lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic hyperbolic composite system in its principal part meanwhile some efficient weighted sobolev estimates are established to derive the existence for corresponding nonlinear problems

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

transonic shocks in compressible flow passing a duct for three dimensional euler systems in this paper we study the transonic shock in steady compressible flow passing a duct the flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front which passes through a given point on the wall of the duct the flow is governed by the three dimensional steady full euler system which is purely hyperbolic ahead of the shock and is of elliptic hyperbolic composed type behind the shock the upstream flow is a uniform supersonic one with the addition of a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem of determining the transonic shock and the flow behind the shock is reduced to a free boundary value problem in order to solve the free boundary problem of the elliptic hyperbolic system one crucial point is to decompose the whole system to a canonical form in which the elliptic part and the hyperbolic part are separated at the level of the principal part due to the complexity of the characteristic varieties for the three dimensional euler system the calculus of symbols is employed to complete the decomposition the new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations which are coupled with the three dimensional euler system

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions