sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

on the invariant measures for the two dimensional euler flow it is proven that the canonical gibbs measure associated with a gas of vortices of intensity converges in the limitn 0 n const to a gaussian measure which is invariant for the two dimensional euler equation

the resolving power of seismic amplitude data an anisotropic inversion migration approach a description of the theory and numerical implementation of a 3 d linearized asymptotic anisotropic inversion method based on the generalized radon transform is given we discuss implementation aspects including 1 the use of various coordinate systems 2 regularization by both spectral and bayesian statistical techniques and 3 the effects of limited acquisition apertures on inversion we give applications of the theory in which well resolved parameter combinations are determined for particular experimental geometries and illustrate the interdependence of parameter and spatial resolutions procedures for evaluating uncertainties in the parameter estimates that result from the inversion are derived and demonstrated

a blowup criterion for the compressible nematic liquid crystal flows in dimension two abstract in this paper we establish a blowup criterion for the two dimensional compressible nematic liquid crystal flows the criterion is given in terms of the density and the gradient of direction field where the later satisfies the serrin type blowup criterion for this result we do not need the initial density to be positive

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

elliptic functions green functions and the mean field equations on tori we show that the green functions on flat tori can have either 3 or 5 critical points only there does not seem to be any direct method to attack this problem instead we have to employ sophisticated non linear partial differential equations to study it we also study the distribution of number of critical points over the moduli space of flat tori through deformations the functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori the general picture is also emerged though some of the necessary technicality is still to de developed 1 introduction and statement of results the study of geometric or analytic problems on two dimensional tori is the same as the study of problems on r 2 with doubly periodic data such situations occur naturally in sciences and mathematics since early days the mathematical foundation of elliptic functions was subsequently devel oped in the 19th century it turns out that these special functions are rather deep objects by themselves tori of different shape may result in very dif ferent behavior of the elliptic functions and their associated objects arith metic on elliptic curves is perhaps the eldest and the most vivid example in this paper we show that this is also the case for certain non linear par tial differential equations indeed researches on doubly periodic problems in mathematical physics or differential equations often restrict the study to rectangular tori for simplicity this leaves the impression that the theory for general tori may resemble much the same way as for the rectangular case however this turns out to be false we will show that the solvability of themean field equation dependson the shape of the green function which in turn depends on the geometry of the tori in an essential way recall that the green function g z w on a flat torus t c z 1 z 2 is the unique function on t t which satisfies zg z w w z 1 t

strong solutions of the navier stokes equations for isentropic compressible fluids we study strong solutions of the isentropic compressible navier stokes equations in a domain r3 we first prove the local existence of unique strong solutions provided that the initial data 0 and u0 satisfy a natural compatibility condition the important point in this paper is that we allow the initial vacuum the initial density may vanish in an open subset of we then prove a new uniqueness result and stability result our results are valid for unbounded domains as well as bounded ones

serrin type blowup criterion for full compressible navier stokes system the authors establish a serrin type blowup criterion for the cauchy problem of the three dimensional full compressible navier stokes system which states that a strong or smooth solution exists globally provided that the velocity satisfies serrin s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded in particular this criterion extends the well known serrin s blowup criterion for the three dimensional incompressible navier stokes equations to the three dimensional full compressible system and is just the same as that of the barotropic case

theory of differential offset continuation i introduce a partial differential equation to describe the process of prestack reflection data transformation in the offset midpoint and time coordinates the equation is proved theoretically to provide correct kinematics and amplitudes on the transformed constant offset sections solving an initial value problem with the proposed equation leads to integral and frequency domain offset continuation operators which reduce to the known forms of dip moveout operators in the case of continuation to zero offset

blowup criterion for viscous baratropic flows with vacuum states we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth strong solutions for the 3 dimensional 3d barotropic compressible navier stokes equations more precisely if a solution of the 3d barotropic compressible navier stokes equations is initially regular and loses its regularity at some later time then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches our result is the same as ponce s criterion for 3 dimensional incompressible euler equations ponce in commun math phys 98 349 353 1985 in addition initial vacuum states are allowed in our cases

computation of unmeasured third generation vct views from measured views we compute unmeasured cone beam projections from projections measured by a third generation helical volumetric computed tomography system by solving a characteristic problem for an ultrahyperbolic differential equation john 1938 by working in the fourier domain we convert the second order pde into a family of first order ordinary differential equations a simple first order integration is used to solve the odes

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

the applicability of dip moveout azimuth moveout in the presence of caustics reflection seismic data continuation is the computation of data at source and receiver locations that differ from those in the original data using whatever data are available we develop a general theory of data continuation in the presence of caustics and illustrate it with three examples dip moveout dmo azimuth moveout amo and offset continuation this theory does not require knowledge of the reflector positions we construct the output data set from the input through the composition of three operators an imaging operator a modeling operator and a restriction operator this results in a single operator that maps directly from the input data to the desired output data we use the calculus of fourier integral operators to develop this theory in the presence of caustics for both dmo and amo we compute impulse responses in a constant velocity model and in a more complicated model in which caustics arise this analysis reveals errors that can be introduced by assuming for example a model with a constant vertical velocity gradient when the true model is laterally heterogeneous data continuation uses as input a subset common offset common angle of the available data which may introduce artifacts in the continued data one could suppress these artifacts by stacking over a neighborhood of input data using a small range of offsets or angles for example we test data continuation on synthetic data from a model known to generate imaging artifacts we show that stacking over input scattering angles suppresses artifacts in the continued data

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

a beale kato majda criterion for three dimensional compressible viscous heat conductive flows we prove a blow up criterion in terms of the upper bound of 1 for a strong solution to three dimensional compressible viscous heat conductive flows the main ingredient of the proof is an a priori estimate for a quantity independently introduced in haspot regularity of weak solutions of the compressible isentropic navier stokes equation arxiv 1001 1581 2010 and sun et al j math pure appl 95 36 47 2011 whose divergence can be viewed as the effective viscous flux

the inversion problem and applications of the generalized radon transform we prove that under certain conditions the inversion problem for the generalized radon transform reduces to solving a fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation we consider applications of the generalized radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential radon transforms

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

microlocal analysis of seismic inverse scattering in anisotropic elastic media seismic data is modeled in the high frequency approximation using the techniques of microlocal analysis we consider general anisotropic elastic media our methods are designed to allow for the formation of caustics the data is modeled in two ways first we give a microlocal treatment of the kirchhoff approximation where the medium is assumed to be piecewise smooth and reflection and transmission occur at interfaces second we give a refined view on the born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium the joint formulation of born and kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters the latter allows the treatment of scattering up to grazing angles the outcome of the analysis is a characterization of the singular part of seismic data we obtain a set of pseudodifferential operators that annihilate the data in the process we construct a fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function in our construction this fourier integral operator becomes invertible we give the conditions for invertibility for general acquisition geometry the result is also of interest for inverse scattering in acoustic media c 2002 john wiley sons inc

sharp existence results for mean field equations with singular data abstract let be a simply connected open and bounded domain in r 2 we are concerned with the nonlinear elliptic problem 0 1 v 8 e v e v 4 j 1 m j p j in v 0 on where j 0 p j denotes the dirac mass with singular point p j and p 1 p m we provide necessary and sufficient conditions for the existence of solutions to 0 1 our result is the two dimensional version of the sharp existence nonexistence result obtained in druet 2002 13 for elliptic equations with critical exponent in dimension 3 in particular we prove that the set m is open where for a given 1 m 0 0 m p 1 p m problem 0 1 has a solution

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

negative temperature states and large scale long lived vortices in two dimensional turbulence we study onsager s theory of large coherent vortices in turbulent flows in the approximation of the point vortex model for two dimensional euler hydrodynamics in the limit of a large number of point vortices with the energy perpair of vortices held fixed we prove that the entropy defined from the microcanonical distribution as a function of the pair specific energy has its maximum at a finite value and thereafter decreases yielding the negative temperature states predicted by onsager we furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy and under regularity assumptions obeys the joyce montgomery mean field equation we also prove that under appropriate conditions the vorticity distribution is the same as that for the canonical distribution a form of equivalence of ensembles we establish a large fluctuation theory for the microcanonical distributions which is based on a level 3 large deviations theory for exchangeable distributions we discuss some implications of that property for the ergodicity requirements to justify onsager s theory and also the theoretical foundations of a recent extension to continuous vorticity fields by r robert and j miller although the theory of two dimensional vortices is of primary interest our proofs actually apply to a very general class of mean field models with long range interactions in arbitrary dimensions

blow up criterions of strong solutions to 3d compressible navier stokes equations with vacuum abstract in the paper we establish a blow up criterion in terms of the integrability of the density for strong solutions to the cauchy problem of compressible isentropic navier stokes equations in r 3 with vacuum under the assumptions on the coefficients of viscosity 29 3 this extends the corresponding results in huang et al 2011 sun et al 2011 20 36 where a blow up criterion in terms of the upper bound of the density was obtained under the condition 7 as a byproduct the restriction 7 in fan et al 2010 sun et al 2011 12 37 is relaxed to 29 3 for the full compressible navier stokes equations by giving a new proof of lemma 3 1 besides we get a blow up criterion in terms of the upper bound of the density and the temperature for strong solutions to the cauchy problem of the full compressible navier stokes equations in r 3 the appearance of vacuum could be allowed this extends the corresponding results in sun et al 2011 37 where a blow up criterion in terms of the upper bound of 1 was obtained without vacuum the effective viscous flux plays a very important role in the proofs

on steady distributions of self attracting clusters under friction and fluctuations we consider the vlasov fokker planek equation with a newtonian attracting potential and study its stationary solutions given by the generalized lane emden equation in a two dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse for a one dimensional model we prove some results on existence uniqueness stability and symmetry breaking of stationary solutions

characterization and source receiver continuation of seismic reflection data in reflection seismology one places sources and receivers on the earth s surface the source generates elastic waves in the subsurface that are reflected where the medium properties stiffness and density vary discontinuously in the field often there are obstructions to collect seismic data for all source receiver pairs desirable or needed for data processing and application of inverse scattering methods typically data are measured on the earth s surface we employ the term data continuation to describe the act of computing data that have not been collected in the field seismic data are commonly modeled by a scattering operator developed in a high frequency single scattering approximation we initially focus on the determination of the range of the forward scattering operator that models the singular part of the data in the mentioned approximation this encompasses the analysis of the properties of and the construction of a minimal elliptic projector that projects a space of distributions on the data acquisition manifold to the range of the mentioned scattering operator this projector can be directly used for the purpose of seismic data continuation and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator we illustrate the data continuation by a numerical example

some considerations on the nonlinear stability of stationary planar euler flows we give sufficient conditions for the nonlinear stability of possibly nonsmooth stationary solutions of the two dimensional euler equation in symmetric bounded domains we use as lyapunov functions first integrals due to the symmetry of the problem moreover we investigate the stability of smooth solutions under perturbations of the boundary the last result is based on a generalization of the well known arnold approach

global strong solution to the 2d nonhomogeneous incompressible mhd system abstract in this paper we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible mhd system as long as the initial data satisfies some compatibility condition as a corollary the global existence of strong solution with vacuum to the 2d nonhomogeneous incompressible navier stokes equations is also established our main result improves all the previous results where the initial density need to be strictly positive the key idea is to use some critical sobolev inequality of logarithmic type which is originally due to brezis and wainger 1980 7

long time behavior of solution for the compressible nematic liquid crystal flows in r 3 abstract in this paper we investigate the global existence and long time behavior of classical solution for the compressible nematic liquid crystal flows in three dimensional whole space first of all the global existence of classical solution is established under the condition that the initial data are close to the constant equilibrium state in h n r 3 n 3 framework then one establishes algebraic time decay for the classical solution by weighted energy method finally the algebraic decay rate of classical solution in l p r 3 norm with 2 p and optimal decay rate of their spatial derivative in l 2 r 3 norm are obtained if the initial perturbation belong to l 1 r 3 additionally

uniform well posedness and low mach number limit to the compressible nematic liquid crystal flows in a bounded domain abstract we prove global in time and uniform in of the strong solutions to the 3 d compressible nematic liquid crystal flows in a bounded domain where is the mach number consequently we obtain the strong solution of compressible nematic liquid crystal model that converges to that of incompressible nematic liquid crystal model this is the first result on the low mach number limit for compressible nematic liquid crystal flows in 3 d bounded domain our proof relies on the dedicated estimates on the solutions and the subtle use of the boundary conditions

on the propagation of polarization sets for systems of real principal type we define and study the polarization sets for vector valued distributions systems of real principal type are defined and we prove that for such systems of pseudodifferential operators we have a unique propagation of the polarization sets for the solutions

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x