quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions on considere le probleme suivant a n dimensions u t au t 0 x u x t 0 t 0 x u x t u x t t 0 x u x 0 0 x ou est un domaine de r n n 2 a est un operateur elliptique lineaire independant du temps est une derivee oblique sur et 1 0 est continument differentiable et croissante et 1

test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

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

mach reflection for the two dimensional burgers equation abstract we study shock reflection for the two 2d burgers equation this model equation is an asymptotic limit of the euler equations and retains many of the features of the full equations a von neumann type analysis shows that the 2d burgers equation has detachment sonic and crocco points in complete analogy with gas dynamics numerical solutions support the detachment sonic criterion for transition from regular to mach reflection there is also strong numerical evidence that the reflected shock in the 2d burgers mach reflection forms a smooth wave near the mach stem as proposed by colella and henderson in their study of the euler equations

diffraction of a weak shock by a wedge we match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock wedge diffraction problem

mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

a linear approximation for the regular reflection of a weak shock at a wedge abstract the problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math lvii 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle between incident shock and wedge is small we can see the weak regular reflection as the disturbance of normal reflection as in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques a delicate a priori estimate and compact arguments we show that the solution of the linearized problem is smooth in the interior and lipschitz continuous up to the degenerate boundary

existence and regularity of solution of mixed boundary value problem of a busemann equation with nonlinear absorb term abstract the busemann equation is a classical equation coming from fluid dynamics the well posed problem and regularity of solution of busemann equation with nonlinear term are interesting and important the busemann equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 with a special nonlinear absorb term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain by means of elliptic regularization technique a delicate prior estimate and compact argument we show that the solution of mixed boundary value problem of busemann equation is smooth in the interior and lipschitz continuous up to the degenerate boundary on some conditions the result is better than the result of classical boundary degenerate elliptic equation

existence and regularity of the solution of a mixed boundary value problem for the keldysh equation with a nonlinear absorption term abstract the keldysh equation is a more general form of the classic tricomi equation from fluid dynamics its well posedness and the regularity of its solution are interesting and important the keldysh equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 adding a special nonlinear absorption term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain similar to the potential fluid shock reflection problem by means of an elliptic regularization technique a delicate a priori estimate and compact argument we show that the solution of a mixed boundary value problem of the keldysh equation is smooth in the interior and lipschitz continuous up to the degenerate boundary under some conditions we believe that this kind of regularity result for the solution will be rather useful

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

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

e h type mach configuration and its stability this paper is devoted to the study of the classification and stability of mach configurations in the mach reflection of a shock wave by a rigid wall the flow behind the reflected shock front can be supersonic or subsonic correspondingly the mach configuration can be classified as e e type and e h type in this paper we proved the stability of e h type mach configurations provided the reflected shock is weak the result is the complement of the stability of e e type mach configurations in our earlier work

a foundation of rough sets theoretical and computational hybrid intelligent system for survival analysis what do we not know about the association between diabetes and survival time our study offers an alternative mathematical framework based on rough sets to analyze medical data and provide epidemiology survival analysis with risk factor diabetes we experiment on three data sets geriatric melanoma and primary biliary cirrhosis a case study reports from 8547 geriatric canadian patients at the dalhousie medical school notification status dead or alive is treated as the censor attribute and the time lived is treated as the survival time the analysis result illustrates diabetes is a very significant risk factor to survival time in our geriatric patients data this paper offers both theoretical and practical guidelines in the construction of a rough sets hybrid intelligent system for the analysis of real world data furthermore we discuss the potential of rough sets artificial neural networks anns and frailty index in predicting survival tendency

a bifurcation problem governed by the boundary condition ii we deal with positive solutions of u a x up in a bounded smooth domain omega subset mathbb r n subject to the boundary condition u v u a parameter p 1 we prove that this problem has a unique positive solution if and only if 0 0 and coincides with the first eigenvalue of an associated eigenvalue problem moreover we find the limit profile of the solution as 1

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

stability of transonic shocks in supersonic flow past a wedge abstract in this paper we study the stability of transonic shocks in steady supersonic flow past a wedge we take the potential flow equation as the mathematical model to describe the compressible flow it is known that in generic case such a problem admits two possible location of shock connecting the flow ahead it and behind it they can be distinguished as supersonic supersonic shock and supersonic subsonic shock or transonic shock both these possible shocks satisfy the rankine hugoniot conditions and entropy condition in this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

parabolic problems with nonlinear absorptions and releases at the boundaries let a p and q be positive constants t d 0 a d be the closure of d w dx 0 t and hu u t u x x this article studies the following heat conduction problem with nonlinear boundary conditions hu 0in w u x 0 u 0 x ond u x 0 t 1 u 0 t p u x a t 1 u a t q 0

a class of quasilinear degenerate elliptic problems abstract we establish the existence of solutions for a class of quasilinear degenerate elliptic equations the equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain and degeneracy only on the boundary equations of transonic gas dynamics for example satisfy this property in the region of subsonic flow and are degenerate across the sonic surface we prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary the maximal rate of blow up at the degenerate boundary is bounded by the degree of degeneracy in the principal coefficients of the quasilinear elliptic operator our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation the pressure gradient equations of the compressible euler equations and the singular quasilinear anisotropic elliptic problems and extend to the class of equations which satisfy the structure condition such as the shallow water equation compressible isentropic two dimensional euler equations and general two dimensional nonlinear wave equations our study provides a general framework to analyze degenerate elliptic problems arising in the self similar reduction of a broad class of two dimensional cauchy problems

the existence and stability of conic shock waves we study the existence of the nonsymmetrical conic shock wave produced by a supersonic flow past a distorted conic projectile for the weak conic shock wave we establish the existence and its linear stability using the mathematical model of an isentropic irrotational flow

two dimensional nonlinear boundary value problems for elliptic equations on considere la regularite limite des solutions du probleme aux valeurs limites non lineaire f x u du d 2 u 0 dans g x u du 0 sur pour des domaines a 2 dimensions f est uniformement elliptique et g depend de du on demontre des estimations de continuite pour des derivees premieres et secondes de u sous des hypotheses faibles pour la regularite de f g et

self similar solutions for weak shock reflection we present numerical solutions of a two dimensional riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the mach reflection of weak shock waves we develop a new numerical scheme to solve the equations in self similar coordinates and use local grid refinement to resolve the solution in the reflection region the solutions contain a remarkably complex structure there is a sequence of triple points and tiny supersonic patches immediately behind the leading triple point that is formed by the reflection of weak shocks and expansion waves between the sonic line and the mach shock an expansion fan originates at each triple point thus resolving the von neumann paradox of weak shock reflection these numerical solutions raise the question of whether there is an infinite sequence of triple points in an inviscid weak shock mach reflection

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

shock diffraction by convex cornered wedges for the nonlinear wave system we are concerned with rigorous mathematical analysis of shock diffraction by two dimensional convex cornered wedges in compressible fluid flow through the nonlinear wave system this shock diffraction problem can be formulated as a boundary value problem for second order nonlinear partial differential equations of mixed elliptic hyperbolic type in an unbounded domain it can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition we establish a global theory of existence and optimal regularity for this shock diffraction problem to achieve this we develop several mathematical ideas and techniques which are also useful for other related problems involving similar analytical difficulties

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

second order elliptic equations with mixed boundary conditions we consider the mixed boundary value problem au f in b0u g0 in b1u g1in where is a bounded open subset of rn whose boundary is divided into disjoint open subsets and by an n 2 dimensional manifold in we assume a is a properly elliptic second order partial differential operator on and bj for j 0 1 is a normal jth order boundary operator satisfying the complementing condition with respect to a on the coefficients of the operators and and are all assumed arbitrarily smooth as announced in bull amer math soc 83 1977 391 393 we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in sobolev spaces in fact we construct an open subset t of the reals such that if ds u hs au 0 then for s 12 mod 1 b0 b1 ds hs 12 hs 32 is a fredholm operator if and only if s t moreover t xew tx where the sets tx are determined algebraically by the coefficients of the operators at x if n 2 tx is the set of all reals not congruent modulo 1 to some exceptional value if n 3 tx is either an open interval of length 1 or is empty and finally if n 4 tx is an open interval of length 1

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

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

the perron process applied to oblique derivative problems abstract the perron process has been used with great success to prove the solvability of the dirichlet problem for linear elliptic equations by elementary methods a modification of this process is used to give an elementary proof of the solvability of the regular oblique derivative problem for linear elliptic equations

existence and stability of perturbed transonic shocks for compressible steady potential flows abstract we establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem we establish gradient estimates and use the fixed point theorem of caristi s which does not require a compactness of the map to establish the existence result we provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region

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

the tricomi problem for a genuinely nonlinear lavrentiev bitsadze equation of mixed type abstract in this paper we consider the tricomi problem for a genuinely nonlinear lavrentiev bitsadze mixed type equation 2 u x 2 sgn y 1 u x 2 2 u y 2 0 whose coefficients depend on the first order derivative of an unknown function we prove the existence of a solution to this problem by a method which can be used to study more difficult problems for nonlinear mixed type equations arising in gas dynamics

uniqueness of stable capillary hypersurfaces in a ball in this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical our result also provides a proof of a conjecture proposed by sternberg and zumbrun j reine angew math 503 63 85 1998 we also prove a heintze karcher ros type inequality for hypersurfaces with free boundary in a ball which together with the new minkowski formula yields a new proof of alexandrov s theorem for embedded cmc hypersurfaces in a ball with free boundary

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

constraints on possible singularities for the unsteady transonic small disturbance utsd equations we discuss the singular behavior of solutions to two dimensional general secondorder uniformly elliptic equations in divergence form with bounded measurable coefficients and discontinuous dirichlet data along a portion of a lipschitz boundary we show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in the boundary data a problem like this arises in the study of self similar solutions to the hyperbolic conservation laws in two space dimensions given by the unsteady transonic small disturbance utsd flow equations these solutions model the reflection of a weak shock wave upon a thin wedge in the regime where the von neumann paradox applies the present result provides a step in the direction of understanding the nature of the solutions to the utsd equations near a triple point it shows that the flow behind the triple point cannot be strictly subsonic under some mild assumptions on the solutions c 1999 john wiley sons inc

subsonic solutions to compressible transonic potential problems for isothermal self similar flows and steady flows abstract we establish boundary and interior gradient estimates and show that no supersonic bubble appears inside of a subsonic region for transonic potential flows for both self similar isothermal and steady problems we establish an existence result for the self similar isothermal problem and improve the hopf maximum principle to show that the flow is strictly elliptic inside of the subsonic region for the steady problem

intermediate schauder theory for second order parabolic equations ii existence uniqueness and regularity on etablit et on demontre des theoremes d existence d unicite et de regularite pour des solutions de problemes aux valeurs limites et initiales paraboliques lineaires dans des espaces de holder ponderes

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

multiplicities a deterministic view of nondeterminism abstract we discuss four important problems in automata theory each of which has three natural variants a deterministic one a nondeterministic one and as an in between case a nondeterministic one with multiplicities our goal is to support a general view that the third variant is much closer to the first one than to the second one

subsonic solutions for compressible transonic potential flows abstract we establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely it has been conjectured by experiments and numerical studies that the self similar multidimensional flow changes its type namely hyperbolic far from the origin supersonic region and elliptic near the origin subsonic region furthermore the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation the nonlinear wave equation and the pressure gradient equation namely the coefficients of the potential equation depend on the gradients while others are independent of the gradients we provide techniques to handle the gradients establish interior and boundary gradient estimates for the potential flow in a convex region and answer the conjecture that is the flow is strictly elliptic and the region is subsonic

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

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

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

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

the mixed problem in l p for some two dimensional lipschitz domains we consider the mixed problem left begin array ll delta u 0 quad rm in omega frac partial u partial nu f n quad rm on rm n u f d quad rm on d end array right in a class of lipschitz graph domains in two dimensions with lipschitz constant at most 1 we suppose the dirichlet data fd has one derivative in lp d of the boundary and the neumann data fn is in lp n we find a p0 1 so that for p in an interval 1 p0 we may find a unique solution to the mixed problem and the gradient of the solution lies in lp

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

boundary value problem for a nonlinear equation of mixed type abstract in this paper we consider the boundary value problem for a nonlinear lavrentiev bitsadze equation of mixed type 2 u x 2 sgn y 1 u x 2 2 u y 2 0 whose coefficients depend on the first order derivatives of unknown function above y 0 and below y 0 the equation becomes the nonlinear elliptic equation and nonlinear hyperbolic equation respectively this is different from the equation studied in chen 2011 9 we prove the existence of solution to the problem by the method which can be used to study more difficult problems for nonlinear equations of mixed type arising in gas dynamics

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

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

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

higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance utsd equations abstract we establish c 2 estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance utsd equation we also present lipschitz estimates near the degenerate dirichlet boundary the sonic boundary for the utsd equation

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

a numerical study of 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 the two parameters a and b which define oblique shock initial data correspond to the slopes of the initial shock waves in the upper half plane for each a and b the three constant states in the upper half plane satisfy the rankine hugoniot conditions across the shocks this leads to a two parameter family of oblique shock interaction problems in this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram our study supplements the related theoretical results and conjectures recently obtained by s canic and b l keyfitz we employ a high resolution numerical method which reveals fine solution structures our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave in

a linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition abstract the problem of shock reflection by a wedge which the flow is dominated by the unsteady potential flow equation is a important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math 57 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle which between incident shock and wedge is small we can see that weak regular reflection as the disturbance of normal reflection as in shuxing chen linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with above viewpoint we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques delicate a priori estimate and compact arguments we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and lipschitz continuous up to the degenerate boundary

identification of a temperature dependent heat conductivity from single boundary measurements considering the identification of a temperature dependent conductivity in a quasilinear elliptic heat equation from single boundary measurements we proof uniqueness in dimensions n ge 2 taking noisy data into account we apply tikhonov regularization in order to overcome the instabilities by using a problem adapted adjoint we give convergence rates under substantially weaker and more realistic conditions than required by the general theory our theory is supported by numerical tests

a global subsonic solution to an interacting transonic shock for the self similar nonlinear wave equation abstract we present the existence of the subsonic solution to a two dimensional riemann problem governed by a self similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle by utilizing the barrier methods and iterative methods we show the well posedness of the transonic shock in the entire subsonic region and thus establish the global solution this result does not rely on any smallness of riemann data

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

a singular multi dimensional piston problem in compressible flow abstract this paper concerns the multi dimensional piston problem which is a special initial boundary value problem of multi dimensional unsteady potential flow equation the problem is defined in a domain bounded by two conical surfaces one of them is shock whose location is also to be determined by introducing self similar coordinates the problem can be reduced to a free boundary value problem of an elliptic equation the existence of the problem is proved by using partial hodograph transformation and nonlinear alternating iteration the result also shows the stability of the structure of shock front in symmetric case under small perturbation

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

stability analysis for semilinear parabolic problems in general unbounded domains abstract we introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain under boundary condition of the oblique derivative type we employ these notions in the stability analysis of semilinear problems some of the properties we derive are new even in the dirichlet or in the whole space cases as an application we show the validity of the hair trigger effect for the fisher kpp equation on general uniformly smooth domains

an elliptic problem arising from the unsteady transonic small disturbance equation abstract we prove a theorem on existence of a weak solution of the dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary the degeneracy is of a type keldysh type associated with singular behavior blow up of a derivative at the boundary we define an associated operator which is continuous pseudo monotone and coercive and show that a weak solution displaying singular behavior at the boundary exists