the steady state collision of two compressible subsonic perfect flows abstract this paper is devoted to the mathematical theory for the steady state collision of two compressible subsonic irrotational flows issuing from two infinitely long nozzles we established the existence uniqueness and asymptotic behavior of the subsonic collision flow with a smooth interface for the steady euler system in an important physical regime more precisely there exists a critical value if the summation of the incoming mass fluxes in the inlets of the nozzles is less than the critical value then there exists a unique smooth subsonic compressible collision flow and furthermore there exists a smooth interface which separates the two non miscible compressible subsonic flows in particular a key observation is that the location of the interface can be determined uniquely by the incoming mass fluxes in the inlets of the nozzles finally we showed some monotonic relationship between the location of the interface and the incoming mass fluxes

compressible subsonic jet flows issuing from a nozzle of arbitrary cross section abstract we are concerned with the well posedness theory of two dimensional compressible subsonic jet flow issuing from a semi infinitely long nozzle of arbitrary cross section given any atmospheric pressure p 0 we show that there exists a critical mass flux m c r depending on p 0 and such that if the incoming mass flux m 0 is less than the critical value then there exists a unique smooth subsonic jet flow issuing from the given nozzle the jet boundary is a free streamline which initiates from the end point of the nozzle smoothly and extends to the infinity one of the key observations in this paper is that the restriction of the incoming mass flux guarantees completely the subsonicity of the compressible jet in the whole flow field which coincides with the observation on the compressible subsonic flows in an infinitely long nozzle without free boundary in 8

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

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

boundary blow up solutions to elliptic systems of competitive type abstract we consider the elliptic system u u p v q v u r v s in where p s 1 q r 0 and r n is a smooth bounded domain subject to different types of dirichlet boundary conditions f u v i u v and sf u v on where 0 under several hypotheses on the parameters p q r s we show existence and nonexistence of positive solutions uniqueness and nonuniqueness we further provide the exact asymptotic behaviour of the solutions and their normal derivatives near some more general related problems are also studied

a second order estimate for blow up solutions of elliptic equations abstract we investigate second term asymptotic behavior of boundary blow up solutions to the problems u b x f u x subject to the singular boundary condition u x in a bounded smooth domain r n b x is a non negative weight function the nonlinearly f is regularly varying at infinity with index 1 that is lim u f u f u for every 0 and the mapping f u u is increasing on 0 the main results show how the mean curvature of the boundary appears in the asymptotic expansion of the solution u x our analysis relies on suitable upper and lower solutions and the karamata regular variation theory

two phase fluids in collision of incompressible inviscid fluids effluxing from two nozzles abstract this paper is devoted to the mathematical theory of two dimensional injection of incompressible irrotational and inviscid fluids issuing from two infinitely long nozzles into a free stream in general there is a free interface with constant jump of the bernoulli constant on it which is different and more difficult than the previous related works physically it is called the collision fluid the main result in this paper is that for given two co axis symmetric infinitely long nozzles imposing the incoming mass fluxes in the two nozzles there exists a unique piecewise smooth collision fluid such that the free interface of the collision fluid is a c 1 curve and the pressure is continuous across the interface as byproducts the asymptotic behaviors the positivity of the vertical velocity monotone relationship between the location of the interface and the incoming mass fluxes are also established

a remark on uniqueness of large solutions for elliptic systems of competitive type abstract we prove that the semilinear system u a x u p v q v b x u r v s in a smooth bounded domain r n has a unique positive solution with the boundary condition u v on provided that p s 1 q r 0 and p 1 s 1 q r 0 the main novelty is imposing a growth on the possibly singular weights a x b x near rather than requiring them to have a precise asymptotic behavior

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

two dimensional cavity flow in an infinitely long channel with non zero vorticity abstract the main object of this paper is to investigate the well posedness theory of the incompressible inviscid cavity flow in an infinitely long channel the flow is governed by two dimensional incompressible steady euler system the main results read that given a mass flux and a constant vorticity in the inlet of the channel firstly we establish the existence and the uniqueness of the incompressible cavity flow in an infinitely long symmetric channel which contains a smooth free surface detaching at the boundary point of the obstacle secondly some fundamental properties such as the asymptotic behaviors of the cavity flow and the free boundary in the upstream and downstream and the positivity of the horizontal velocity are also obtained finally we show that there does not exist a finite cavity or a cusped cavity in the infinitely long nozzle which gives a positive answer to the conjecture by h villat in 1913 on the non existence of a symmetric finite cusped cavity behind an obstacle

the best test of exponentiality against singly truncated normal alternatives abstract we show that the likelihood ratio test of exponentiality against singly truncated normal alternatives is the uniformly most powerful unbiased test and can be expressed in terms of the sampling coefficient of variation this test is closely related to greenwood s statistic for testing departures from the uniform distribution we provide a way to approximate the critical points of the test using saddlepoint methods that gives a high degree of accuracy

boundary behaviors for liouville s equation in planar singular domains abstract we study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary the conformal structure plays an essential role

scale invariance and universality organizing principles in complex systems this paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the eld of phase transitions and critical phenomena scale invariance and universality can be useful in guiding research on a broad class of complex phenomena we shall see that while scale invariance has been tested for many years universality is relatively more rarely discussed in particular we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems c 2000 elsevier science b v all rights reserved

boundary asymptotic and uniqueness of solutions to the p laplacian with infinite boundary values we prove boundary asymptotics to solutions of weighted p laplacian equations that take infinite value on the boundary of a bounded domain uniqueness of such solutions would then follow as a consequence our results extend previously known results by allowing weights that are unbounded in the domain

asymptotic estimates of boundary blow up solutions to the infinity laplace equations abstract in this paper we study the asymptotic behavior of boundary blow up solutions to the equation u b x f u in where is the laplacian the nonlinearity f is a positive increasing function in 0 and the weighted function b c is positive in and may vanish on the boundary we first establish the exact boundary blow up estimates with the first expansion when f is regularly varying at infinity with index p 3 and the weighted function b is controlled on the boundary in some manner furthermore for the case of f s s p 1 c g s with the function g normalized regularly varying with index q 0 we obtain the second expansion of solutions near the boundary it is interesting that the second term in the asymptotic expansion of boundary blow up solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the boundary blow up problems involving the classical laplacian

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

subsonic non isentropic euler flows with large vorticity in axisymmetric nozzles abstract in this paper we investigate the effects of vorticity for smooth subsonic non isentropic euler flows in infinitely long axisymmetric nozzles it is a sequel to b duan and z luo 2013 13 on non isentropic euler system in 13 the authors obtained the unique existence of a smooth subsonic solution with restriction on the smallness of bernoulli s function in the upstream the smallness assumption is sufficient to prevent the appearance of singularity however it leads that the vorticity is also small the purpose of this paper is to remove such restriction on the smallness of the vorticity

global subsonic and subsonic sonic flows through infinitely long axially symmetric nozzles abstract in this paper we establish existence of global subsonic and subsonic sonic flows through infinitely long axially symmetric nozzles by combining variational method various elliptic estimates and a compensated compactness method more precisely it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value moreover uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields and flow angles for axially symmetric flows are uniformly bounded away from 2 finally when the incoming mass flux tends to the critical value subsonic sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework

the second order expansion of boundary blow up solutions for infinity laplacian equations abstract in this paper by using karamata regular variation theory and a perturbed argument we study the second order expansion of viscosity solutions to boundary blow up elliptic problem u b x f u x u where is a bounded domain with c 2 boundary in r n b c is non negative and non trivial in f c 1 0 is a normalized regularly varying function at infinity with index 3

two dimensional subsonic euler flows past a wall or a symmetric body the existence and uniqueness of two dimensional steady compressible euler flows past a wall or a symmetric body are established more precisely given positive convex horizontal velocity in the upstream there exists a critical value rho rm cr such that if the incoming density in the upstream is larger than rho rm cr then there exists a subsonic flow past a wall furthermore rho rm cr is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than rho rm cr the subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary moreover the existence and uniqueness of a two dimensional subsonic euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value the asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states

exact multiplicity for boundary blow up solutions abstract the singularly perturbed boundary blow up problem e 2 u u u a 1 u u 0 in b u on b is studied in the unit ball b r n n 2 a 1 2 1 is a constant it is shown that for e 0 sufficiently small there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions

blow up rates of large solutions for elliptic equations in this paper we mainly study the boundary behavior of solutions to boundary blow up elliptic problems for more general nonlinearities f which may be rapidly varying at infinity u b x f u x u where is a bounded domain with smooth boundary in rn and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary further when f s sp f1 s for s sufficiently large where p 1 and f1 is normalized regularly varying at infinity with index p1 0 p we show the influence of the geometry of on the boundary behavior for solutions to the problem we also give the existence and uniqueness of solutions

global subsonic euler flows in an infinitely long axisymmetric nozzle abstract in this paper we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle the flow is governed by the steady euler equations and has boundary conditions on the nozzle walls existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle when the variation of bernoulli s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value the results give a strictly mathematical proof to the assertion in bers 1958 2 there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle provided that the incoming mass flux is less than the critical value the existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables with the assumptions on the nozzle in the far fields the asymptotic behavior can be derived by a blow up argument for the infinitely long nozzle finally we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux

asymptotic boundary estimates to infinity laplace equations with γ varying nonlinearity abstract in a previous paper of the authors wang et al 2014 40 the asymptotic estimates of boundary blow up solutions were established to the infinity laplace equation u b x f u in r n with the nonlinearity 0 f c 0 regularly varying at and the weighted function b c positive in and vanishing on the boundary the present paper gives a further investigation on the asymptotic behavior of boundary blow up solutions to the same equation by assuming f to be varying note that a varying function grows faster than any regularly varying function we first quantitatively determine the boundary blow up estimates with the first expansion relying on the decay rate of b near the boundary and the growth rate of f at infinity and further characterize these results via examples possessing various decay rates for b and growth rates for f in particular we pay attention to the second order estimates of boundary blow up solutions it was observed in our previous work that the second expansion of solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the classical laplacian the second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow up solutions between the infinity laplacian and the classical laplacian

on two dimensional sonic subsonic flow a compensated compactness framework is established for sonic subsonic approximate solutions to the two dimensional euler equations for steady irrotational flows that may contain stagnation points only crude estimates are required for establishing compactness it follows that the set of subsonic irrotational solutions to the euler equations is compact thus flows with sonic points over an obstacle such as an airfoil may be realized as limits of sequences of strictly subsonic flows furthermore sonic subsonic flows may be constructed from approximate solutions the compactness framework is then extended to self similar solutions of the euler equations for unsteady irrotational flows

subsonic sonic limit of approximate solutions to multidimensional steady euler equations a compactness framework is established for approximate solutions to subsonic sonic flows governed by the steady full euler equations for compressible fluids in arbitrary dimension the existing compactness frameworks for the two dimensional irrotational case do not directly apply for the steady full euler equations in higher dimensions the new compactness framework we develop applies for both non homentropic and rotational flows one of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity along with the bernoulli law and the entropy relation through a more delicate analysis on the phase space as direct applications we establish two existence theorems for multidimensional subsonic sonic full euler flows through infinitely long nozzles

nondegeneracy and uniqueness for boundary blow up elliptic problems abstract in this paper we use for the first time linearization techniques to deal with boundary blow up elliptic problems after introducing a convenient functional setting we show that the problem u a x u p g x u in with u on has a unique positive solution for large enough and determine its asymptotic behavior as here p 1 a x is a continuous function which can be singular near and g x u is a perturbation term with potential growth near zero and infinity we also consider more general problems obtained by replacing u p by e u or a logistic type function f u

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

large solutions for a class of semilinear integro differential equations with censored jumps we study existence of large solutions that is solutions that verify u x 1 as x for equations like i u x u x p 0 x2 where is a bounded smooth domain in r n p 1 andi is a nonlocal operator of the form i u x p v z jzj x u x z u x jzj n dz where 2 0 2 and r is a function whose main particularity is that 0 x dist x we also obtain uniqueness of the solution in a class of large solutions whose blow up rate depends on p and the rate at which shrinks near the boundary

steady subsonic ideal flows through an infinitely long nozzle with large vorticity in this paper the existence uniqueness and far field behavior of a class of subsonic flows with large vorticity for the steady euler equations in infinitely long nozzles are established more precisely for any given convex horizontal velocity of incoming flow in the upstream there exists a critical value mcr if the mass flux is larger than mcr then there exists a unique smooth subsonic euler flow through the infinitely long nozzle this well posedness result is proved by a new observation for the method developed in xie and xin siam j math anal 42 751 784 2010 to deal with the euler system furthermore the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions

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

second order estimates for large solutions of elliptic equations abstract this paper deals with the second term asymptotic behavior of large solutions to the problems u b x f u x subject to the singular boundary condition u x x where is a smooth bounded domain in r n and b x is a non negative weight function the absorption term f is regularly varying at infinite with index 1 that is lim u f u f u for every 0 and the mapping f u u is increasing on 0 our analysis relies on the karamata regular variation theory

the second expansion of large solutions for semilinear elliptic equations abstract in this paper we analyze the influence of the domain geometry in the second expansion of solutions to the boundary blow up elliptic problem u b x f u u 0 x u where is a bounded domain with c 4 smooth boundary in r n b c which is positive in and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p p 1

on incompressible oblique impinging jet flows abstract the object of this work is to investigate the fluid mechanics of oblique impinging jet flows and to this end some existence and nonexistence results are initiated first we established the existence of incompressible oblique impinging jet plane flows with two asymptotic directions more precisely given a two dimensional semi infinitely long nozzle and a wall behind the nozzle for any given mass flux in the inlet of the nozzle then there exists a smooth incompressible oblique impinging jet flow with two asymptotic directions the impinging jet develops two free streamlines which initiate smoothly at the endpoints of the semi infinitely long nozzle and the speed on free streamlines remains a constant which can be determined by the impinging jet flow itself the asymptotic behaviors of the oblique impinging jet flows at the far fields the position of the stagnation points convexity of the free boundaries and other properties are also considered on another side it is showed in this paper that there does not exist an oblique impinging jet flow with one asymptotic direction generally

three dimensional full euler flows in axisymmetric nozzles abstract in this paper we consider global subsonic compressible flows through general infinitely long nozzles the flow is governed by the steady full euler equations we find a new invariant along streamline take advantages of this we reformulate the full euler system into quasilinear second order equations of stream function with the upstream condition and the solid wall boundary condition we obtain the existence uniqueness asymptotic behavior and the critical mass flux

universality of power law exponents by means of maximum likelihood estimation power law type distributions are extensively found when studying the behavior of many complex systems however due to limitations in data acquisition empirical datasets often only cover a narrow range of observation making it difficult to establish power law behavior unambiguously in this work we present a statistical procedure to merge different datasets with two different aims first we obtain a broader fitting range for the statistics of different experiments or observations of the same system second we establish whether two or more different systems may belong to the same universality class by means of maximum likelihood estimation this methodology provides rigorous statistical information to discern whether power law exponents characterizing different datasets can be considered equal among them or not this procedure is applied to the gutenberg richter law for earthquakes and for synthetic earthquakes acoustic emission events generated in the laboratory labquakes different earthquake catalogs have been merged finding a gutenberg richter law holding for more than eight orders of magnitude in seismic moment the value of the exponent of the energy distribution of labquakes depends on the material used in the compression experiments by means of the procedure proposed in this manuscript we find that the gutenberg richter law for earthquakes and charcoal labquakes can be characterized by the same power law exponent whereas vycor labquakes exhibit a significantly different exponent

expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation the main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation delta u lambda u m f in omega where lambda 0 m 1 f is an element of c omega f 0 and omega is an open bounded set of r n n 1 with boundary smooth enough roughly speaking we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite but it goes to infinity as in goes to 1 we prove that the expansion consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial derivative omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function dist x partial derivative omega for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function rm dist x partial omega dist x for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources

testing the pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities fat tail distributions of sizes abound in natural physical economic and social systems the lognormal and the power laws have historically competed for recognition with sometimes closely related generating processes and hard to distinguish tail properties this state of affair is illustrated with the debate between eeckhout amer econ rev 94 1429 2004 and levy amer econ rev 99 1672 2009 on the validity of zipf s law for us city sizes by using a uniformly most powerful unbiased umpu test between the lognormal and the power laws we show that conclusive results can be achieved to end this debate we advocate the umpu test as a systematic tool to address similar controversies in the literature of many disciplines involving power laws scaling fat or heavy tails in order to demonstrate that our procedure works for data sets other than the us city size distribution we also briefly present the results obtained for the power law tail of the distribution of personal identity id losses which constitute one of the major emergent risks at the interface between cyberspace and reality

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

large solutions of elliptic semilinear equations in the borderline case an exhaustive and intrinsic point of view abstract we revisit the problem u f u in u x as x where r n n 1 is a bounded smooth domain and f is an increasing and continuous function in r with f 0 0 for which the keller osserman condition holds we study uniqueness of solutions extending known results about the boundary blow up behavior of solutions furthermore we obtain explicit representations for the second order terms in the explosive boundary expansion of solutions under intrinsic and direct assumptions our study is exhaustive including both ordinary and borderline cases providing new and sharp results

boundary behavior for large solutions to elliptic equations with singular weights abstract in this paper we analyze the boundary behavior of large positive solutions to some semilinear elliptic equations which include a singular weight the most important point is that the growth of the solutions can be determined in terms of the solution to a one dimensional first order equation we also consider the questions of existence and uniqueness of positive solutions

symmetry results of large solutions for semilinear cooperative elliptic systems abstract in this paper we consider symmetry of large solutions of the semilinear cooperative elliptic system u i g i u 1 u m in b r u i on b r where u i c 2 b r g i r m r a c function b r is the open ball of center 0 and radius r 0 in r n n 3 m n i 1 2 m after introducing the weakly coupled fully coupled hypotheses and some assumptions for g we use the moving plane method to prove any large solution of the semilinear cooperative elliptic system is radially symmetric and radially increasing in order to use the moving plane method for our problem a strong maximum principle of the linear cooperative elliptic system is constructed furthermore we also get the symmetry result of large solutions of the semilinear cooperative elliptic system in an annulus

structural stability of subsonic irrotational flows in two dimensional infinitely long nozzles abstract this paper investigates the structural stability of two dimensional subsonic irrotational inviscid flows throughout infinitely long nozzles more precisely we show that the smooth subsonic potential flows are structural stable under the small perturbations for the nozzle walls the analysis of this paper is based on the standard estimates for second order quasilinear elliptic equations for the stream function

asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth this paper concerns the blow up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights specifically we consider radially symmetric solutions of mu c x u p on an annulus x rn x with 0 and 0 that diverge to infinity as x here n m n p 1 and c is a positive continuous function on the interval letting r qc r q for r with q 2m p 1 q q q 1 q 2m 1 1 p 1 and c c 1 p 1 we show that as x the ratio u x x remains between positive constants that depend only on m and p extending well known results for the second order problem we prove in the fourth order case that u x x 1 as x and obtain precise asymptotic expansions if c is sufficiently smooth at in certain higher order cases we find solutions for which the ratio u x x does not converge but oscillates about 1 with non vanishing amplitude

contact discontinuities for 3 d axisymmetric inviscid compressible flows in infinitely long cylinders abstract we prove the existence of a subsonic axisymmetric weak solution u p with u u x e x u r e r u e to steady euler system in a three dimensional infinitely long cylinder n when prescribing the values of the entropy p and angular momentum density r u at the entrance by piecewise c 2 functions with a discontinuity on a curve on the entrance of n due to the variable entropy and angular momentum density swirl conditions with a discontinuity at the entrance the corresponding solution has a nonzero vorticity nonzero swirl and contains a contact discontinuity r g d x we construct such a solution via helmholtz decomposition the key step is to decompose the rankine hugoniot conditions on the contact discontinuity via helmholtz decomposition so that the compactness of approximated solutions can be achieved then we apply the method of iteration to obtain a solution and analyze the asymptotic behavior of the solution at far field

subsonic flows in a multi dimensional nozzle in this paper we study the global subsonic irrotational flows in a multi dimensional n 2 infinitely long nozzle with variable cross sections the flow is described by the inviscid potential equation which is a second order quasilinear elliptic equation when the flow is subsonic first we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension with sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow then we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely provided that the incoming mass flux is less than the critical value this gives a positive answer to the problem of bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension bers in surveys in applied mathematics vol 3 wiley new york 1958 finally under suitable asymptotic assumptions of the nozzle we obtain the asymptotic behavior of the subsonic flow in far fields by means of a blow up argument the main ingredients of our analysis are methods of calculus of variations the moser iteration techniques for the potential equation and a blow up argument for infinitely long nozzles

second order expansion for blowup solutions of semilinear elliptic problems abstract in this paper we mainly study the second order expansion of solutions to the elliptic problems u b x f u u 0 x u where is a bounded domain with a smooth boundary in r n n 2 b c which is positive in and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p 1 our analysis is based on the sub supersolution method and karamata regular variation theory

steady compressible subsonic impinging flows with non zero vorticity abstract a well posedness analysis of steady state of inviscid compressible subsonic impinging flows with non zero vorticity is developed a nonlinear nonhomogeneous second order partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specifying the horizontal velocity we first construct a subsonic solution to the impinging flow problem with special horizontal velocity and specific effluent fluxes in the outlets moreover we showed that such a subsonic impinging flow with the same asymptotic behavior in the upstream and downstreams is unique furthermore the existence uniqueness and regularity of the interface separating the two fluids with different outlets are obtained this result extends the recent result on the compressible subsonic irrotational impinging flows in 5 finally as a byproduct we obtain the existence of subsonic sonic impinging flows