phase change problems with temperature dependent constraints for the volume fraction velocities abstract a class of nonlinear systems of parabolic pdes is considered as a mathematical model for phase change phenomena arising in binary mixtures we use the volume fraction of one of the components as an order parameter in our systems therefore our system consists of two kinetic equations for the internal energy and order parameter which are derived in the non smooth thermomechanics theory in this paper we give a general treatment for our system and establish an existence result by applying the schauder s fixed point theory with the subdifferential operator techniques

singular behavior of vacuum states for compressible fluids in this survey paper we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics that is euler equations euler poisson equations and navier stokes equations the main concern is the well posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas furthermore the relation of the solutions for the gas dynamics with vacuum to those of the boltzmann equation will also be discussed in fact the results obtained so far for vacuum states are far from being complete and satisfactory therefore this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems moreover the problems considered here are limited to the author s interest and knowledge in this area

when is the shape of a scene unique given its light field a fundamental theorem of 3d vision the complete set of measurements that could ever be used by a passive 3d vision algorithm is the plenoptic function or light field we give a concise characterization of when the light field of a lambertian scene uniquely determines its shape and conversely when the shape is inherently ambiguous in particular we show that stereo computed from the light field is ambiguous if and only if the scene is radiating light of a constant intensity and color etc over an extended region

epipolar plane image analysis an approach to determining structure from motion we present a technique for building a three dimensional description of a static scene from a dense sequence of images these images are taken in such rapid succession that they form a solid block of data in which the temporal continuity from image to image is approximately equal to the spatial continuity in an individual image the technique utilizes knowledge of the camera motion to form and analyze slices of this solid these slices directly encode not only the three dimensional positions of objects but also such spatiotemporal events as the occlusion of one object by another for straight line camera motions these slices have a simple linear structure that makes them easier to analyze the analysis computes the three dimensional positions of object features marks occlusion boundaries on the objects and builds a three dimensional map of free space in our article we first describe the application of this technique to a simple camera motion and then show how projective duality is used to extend the analysis to a wider class of camera motions and object types that include curved and moving objects

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

spacetime stereo a unifying framework for depth from triangulation depth from triangulation has traditionally been investigated in a number of independent threads of research with methods such as stereo laser scanning and coded structured light considered separately we propose a common framework called spacetime stereo that unifies and generalizes many of these previous methods to show the practical utility of the framework we develop two new algorithms for depth estimation depth from unstructured illumination change and depth estimation in dynamic scenes based on our analysis we show that methods derived from the spacetime stereo framework can be used to recover depth in situations in which existing methods perform poorly

a theory of shape by space carving in this paper we consider the problem of computing the 3d shape of an unknown arbitrarily shaped scene from multiple photographs taken at known but arbitrarily distributed viewpoints by studying the equivalence class of all 3d shapes that reproduce the input photographs we prove the existence of a special member of this class the photo hull that 1 can be computed directly from photographs of the scene and 2 subsumes all other members of this class we then give a provably correct algorithm called space carving for computing this shape and present experimental results on complex real world scenes the approach is designed to 1 capture photorealistic shapes that accurately model scene appearance from a wide range of viewpoints and 2 account for the complex interactions between occlusion parallax shading and their view dependent effects on scene appearance

solutions of euler poisson equations for gaseous stars in this paper we study the euler poisson equations governing gas motion under self gravitational force we are interested in the evolution of the gaseous stars for which the density function has compact support we establish existence theory for the stationary solutions and describe the behavior of the solutions near the vacuum boundary the boundary behavior thus obtained agrees with the physical boundary condition proposed and studied in l ly for both euler equations with damping and the euler poisson equations existence non existence uniqueness and instability of the stationary solutions with vacuum are also discussed in terms of the adiabatic exponent and the entropy function and the phenomena of the blowup that is the collapsing of the star to a single point with finite mass as well as the drifting of part of the star to infinity in space are also studied and shown to agree with the conjecture from the physical considerations

well posedness for the motion of a compressible liquid with free surface boundary we study the motion of a compressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

existence and non linear stability of rotating star solutions of the compressible euler poisson equations we prove the existence of rotating star solutions which are steady state solutions of the compressible isentropic euler poisson euler poisson equations in three spatial dimensions with prescribed angular momentum and total mass this problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical auchmuty beals paper we prove the non linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution we also prove finite time stability of w 1 infty mathbb r 3 solutions where the perturbations are entropy weak solutions of the euler poisson equations finally we give a uniform in time a priori estimate for entropy weak solutions of the euler poisson equations

compressible euler equations with vacuum abstract in this paper we first show that the regular solutions of compressible euler equations in r 3 with damping will not be global if the initial density function has compact support this implies that c 2 cannot be smooth across the boundary separating the gas and the vacuum after a finite time where c is the speed of sound then we study the local existence of solutions for isentropic gas flow in r when c 0 1 is smooth across using the energy method and the characteristics method

variational principles surface evolution pdes level set methods and the stereo problem we present a novel geometric approach for solving the stereo problem for an arbitrary number of images spl ges 2 it is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images the euler lagrange equations that are deduced from the variational principle provide a set of partial differential equations pdes that are used to deform an initial set of surfaces which then move toward the objects to be detected the level set implementation of these pdes potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformations i e to deal with multiple objects results of an implementation of our theory also dealing with occlusion and visibility are presented on synthetic and real images

a multiple baseline stereo a stereo matching method that uses multiple stereo pairs with various baselines generated by a lateral displacement of a camera to obtain precise distance estimates without suffering from ambiguity is presented matching is performed simply by computing the sum of squared difference ssd values the ssd functions for individual stereo pairs are represented with respect to the inverse distance and are then added to produce the sum of ssds this resulting function is called the sssd in inverse distance it is shown that the sssd in inverse distance function exhibits a unique and clear minimum at the correct matching position even when the underlying intensity patterns of the scene include ambiguities or repetitive patterns the authors first define a stereo algorithm based on the sssd in inverse distance and present a mathematical analysis to show how the algorithm can remove ambiguity and increase precision experimental results with real stereo images are presented to demonstrate the effectiveness of the algorithm

quasivariational evolution inequalities for a class of reaction diffusion systems abstract this paper is concerned with a class of doubly nonlinear quasivariational evolution inequalities for a class of reaction diffusion models with time derivative constraint we prove an existence result by using the fixed point argument with the monotonicity technique in the nonlinear operator theory

unidirectional evolution equations of diffusion type abstract this paper is concerned with the uniqueness existence partial smoothing effect comparison principle and long time behavior of solutions to the initial boundary value problem for a unidirectional diffusion equation the unidirectional evolution often appears in damage mechanics due to the strong irreversibility of crack propagation or damage evolution the existence of solutions is proved in an l 2 framework by employing a backward euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems the novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long time behavior of solutions

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

the visual hull concept for silhouette based image understanding many algorithms for both identifying and reconstructing a 3 d object are based on the 2 d silhouettes of the object in general identifying a nonconvex object using a silhouette based approach implies neglecting some features of its surface as identification clues the same features cannot be reconstructed by volume intersection techniques using multiple silhouettes of the object this paper addresses the problem of finding which parts of a nonconvex object are relevant for silhouette based image understanding for this purpose the geometric concept of visual hull of a 3 d object is introduced this is the closest approximation of object s that can be obtained with the volume intersection approach it is the maximal object silhouette equivalent to s i e which can be substituted for s without affecting any silhouette only the parts of the surface of s that also lie on the surface of the visual hull can be reconstructed or identified using silhouette based algorithms the visual hull depends not only on the object but also on the region allowed to the viewpoint two main viewing regions result in the external and internal visual hull in the former case the viewing region is related to the convex hull of s in the latter it is bounded by s the internal visual hull also admits an interpretation not related to silhouettes algorithms for computing visual hulls are presented and their complexity analyzed in general the visual hull of a 3 d planar face object turns out to be bounded by planar and curved patches

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

local existence with physical vacuum boundary condition to euler equations with damping abstract in this paper we consider the local existence of solutions to euler equations with linear damping under the assumption of physical vacuum boundary condition by using the transformation introduced in lin and yang methods appl anal 7 3 2000 495 to capture the singularity of the boundary we prove a local existence theorem on a perturbation of a planar wave solution by using littlewood paley theory and justifies the transformation introduced in liu and yang 2000 in a rigorous setting

well posedness for the motion of physical vacuum of the three dimensional compressible euler equations with or without self gravitation this paper concerns the well posedness theory of the motion of a physical vacuum for the compressible euler equations with or without self gravitation first a general uniqueness theorem of classical solutions is proved for the three dimensional general motion second for the spherically symmetric motions without imposing the compatibility condition of the first derivative being zero at the center of symmetry a new local in time existence theory is established in a functional space involving less derivatives than those constructed for three dimensional motions in coutand et al commun math phys 296 559 587 2010 coutand and shkoller arch ration mech anal 206 515 616 2012 jang and masmoudi well posedness of compressible euler equations in a physical vacuum 2008 by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary