growth rates for the linearized motion of fluid interfaces away from equilibrium we consider the motion of a two dimensional interface separating an inviscid incompressible irrotational fluid influenced by gravity from a region of zero density we show that under certain conditions the equations of motion linearized about a presumed time dependent solution are wellposed that is linear disturbances have a bounded rate of growth if surface tension is neglected the linear equations are well posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity similar to the criterion formulated by g i taylor if surface tension is included the linear equations are well posed without qualifications whether the fluid is above or below the interface an interesting qualitative structure is found for the linear equations a lagrangian approach is used like that of numerical work such as 3 except that the interface is assumed horizontal at infinity certain integral equations which occur involving double layer potentials are shown to be solvable in the present case 1993 john wiley sons inc

boundary layers associated with incompressible navier stokes equations the noncharacteristic boundary case abstract the goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic i e there is injection and or suction everywhere at the boundary following earlier work on the boundary layer for linearized navier stokes equations in the case where the boundaries are characteristic no slip at the boundary and non permeable we consider here the case where the boundary is permeable and thus noncharacteristic the form of the boundary layer and convergence results are derived in two cases linearized equation and full nonlinear equations we prove that there exists a boundary layer at the outlet downwind of the form e uz e where u is the speed of injection suction at the boundary z is the distance to the outlet of the channel and e is the kinematic viscosity we improve an earlier result of s n alekseenko 1994 siberian math j 35 no 2 209 230 where the convergence in l 2 of the solutions of the navier stokes equations to that of the euler equations at vanishing viscosity was established in the two dimensional case we are able to derive the physically relevant uniform in space l norm estimates of the boundary layer the uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic sobolev imbedding to the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full nonlinear navier stokes equations for incompressible fluids

characteristic boundary layers in real vanishing viscosity limits abstract in this paper we study the vanishing viscosity limit of initial boundary value problems for one dimensional mixed hyperbolic parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems in particular we assume that an eigenvalue of the inviscid system vanishes uniformly we prove the stability of boundary layers expansions in small time i e before shocks for the inviscid system as long as the amplitude of the boundary layers remains sufficiently small in particular by using lagrangian coordinates we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous dirichlet condition for the velocity at the boundary

contractions and deformations suppose that f is a projective birational morphism with at most one dimensional fibres between d dimensional varieties x and y satisfying rf ox oy consider the locus l in y over which f is not an isomorphism taking the scheme theoretic fibre c over any closed point of l we construct algebras afib and acon which prorepresent the functors of commutative deformations of c and noncommutative deformations of the reduced fibre respectively our main theorem is that the algebras acon recover l and in general the commutative deformations of neither c nor the reduced fibre can do this as the d 3 special case this proves the following contraction theorem in a neighbourhood of the point the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable

the rigorous approximation of long wavelength capillary gravity waves in a previous paper we proved that long wavelength solutions of the water wave problem in the case of zero surface tension split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries kdv equation in this paper we examine the effect of surface tension on this scenario we find that we obtain three different physical regimes depending on the strength of the surface tension for weak surface tension the propagation of the wave packets is very similar to that in the zero surface tension case for strong surface tension the evolution is again governed by a pair of kdv equations but the coefficients in these equations have changed in such a way that the kdv soliton now represents a wave of depression on the fluid surface finally at a special intermediate value of the surface tension where the bond number equals the kdv description breaks down and it is necessary to introduce a new approximating equation the kawahara equation which is a fifth order nonlinear partial differential equation in each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water wave problem

zero viscosity limit of the navier stokes equations in the analytic setting in this paper we consider the zero viscosity limit of the navier stokes equations in a half space with non slip boundary condition based on the vorticity formulation and the use of conormal derivative we develop an energy method to justify the zero viscosity limit for the analytic data

blowup of solutions of the unsteady prandtl s equation we prove that for certain class of compactly supported c initial data smooth solutions of the unsteady prandtl s equation blow up in nite time

picard groups for derived module categories we introduce in this paper a generalization of picard groups to derived cat egories of algebras we study general properties of those groups and we construct braid group actions on these groups for particular classes of algebras

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

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

zero surface tension limit of the free surface incompressible euler equations with damping abstract we consider the free boundary problem for a layer of an incompressible inviscid fluid in a uniform gravitational field lying above a rigid bottom and below the atmosphere in a horizontally periodic setting we establish the global in time zero surface tension limit of the problem with damping for the small initial data

singular derived categories of factorial terminalizations and maximal modification algebras abstract let x be a gorenstein normal 3 fold satisfying elf with local rings which are at worst isolated hypersurface e g terminal singularities by using the singular derived category d sg x and its idempotent completion d sg x we give necessary and sufficient categorical conditions for x to be q factorial and complete locally q factorial respectively we then relate this information to maximal modification algebras mmas introduced in 20 by showing that if an algebra is derived equivalent to x as above then x is q factorial if and only if is an mma thus all rings derived equivalent to q factorial terminalizations in dimension three are mmas as an application we extend some of the algebraic results in 6 and 14 using geometric arguments

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

on the cauchy problem for gravity water waves we are interested in the system of gravity water waves equations without surface tension our purpose is to study the optimal regularity thresholds for the initial conditions in terms of sobolev embeddings the initial surfaces we consider turn out to be only of c 3 2 epsilon class for some epsilon 0 and consequently have unbounded curvature while the initial velocities are only lipschitz we reduce the system using a paradifferential approach

well posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition abstract we consider the free boundary problem for the flow of an incompressible inviscid elastic fluid the columns of the deformation gradient are tangent and the pressure vanishes along the free interface we prove the local existence of a unique smooth solution to the free boundary problem under the mixed type stability condition provided that the rayleigh taylor sign condition is satisfied at all the points of the initial interface where the non collinearity condition for the deformation gradient among three columns of the deformation gradient there are two non collinear vectors fails in particular we solve an open problem proposed by y trakhinin in the paper 29 under the incompressible setting

gopakumar vafa invariants do not determine flops two 3 fold flops are exhibited both of which have precisely one flopping curve one of the two flops is new and is distinct from all known algebraic d 4 flops it is shown that the two flops are neither algebraically nor analytically isomorphic yet their curve counting gopakumar vafa invariants are the same we further show that the contraction algebras associated to both are not isomorphic so the flops are distinguished at this level this shows that the contraction algebra is a finer invariant than various curve counting theories and it also provides more evidence for the proposed analytic classification of 3 fold flops via contraction algebras

universal behavior of coupled order parameters below three dimensions we explore universal critical behavior in models with two competing order parameters and an mathrm o n ensuremath bigoplus phantom rule 4pt 0ex mathrm o m symmetry for dimensions d ensuremath le 3 in d 3 there is always exactly one stable renormalization group fixed point corresponding to bicritical or tetracritical behavior employing pseudospectral techniques to solve functional renormalization group equations in a two dimensional field space we uncover a more intricate structure of fixed points in dl3 where two additional bicritical fixed points play a role towards d 2 we discover ranges of n m with several simultaneously stable fixed points indicating the coexistence of several universality classes

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

a note on prandtl boundary layers this note concerns nonlinear ill posedness of the prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary our analysis is built upon recent remarkable linear illposedness results established by gerard varet and dormy and an analysis by guo and tice we show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in sobolev spaces we also introduce a notion of weak well posedness and prove that the nonlinear prandtl equation is not well posed in this sense near nonstationary and nonmonotonic shear flows on the other hand we are able to verify that oleinik s monotonic solutions are well posed 2011 wiley periodicals inc

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

noncommutative enhancements of contractions abstract given a contraction of a variety x to a base y we enhance the locus in y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings d under mild assumptions which hold in the case of 1 crepant partial resolutions admitting a tilting bundle with trivial summand and 2 all contractions with fibre dimension at most one in all cases this produces a global invariant in the crepant setting we then apply this to study derived autoequivalences of x we work generally dropping many of the usual restrictions and so both extend and unify existing approaches in full generality we construct a new endofunctor of the derived category of x by twisting over d and then under appropriate restrictions on singularities give conditions for when it is an autoequivalence we show that these conditions hold automatically when the non isomorphism locus in y has codimension 3 or more which covers determinantal flops and we also control the conditions when the non isomorphism locus has codimension 2 which covers 3 fold divisor to curve contractions

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

existence and stability of noncharacteristic boundary layers for the compressible navier stokes and viscous mhd equations for a general class of hyperbolic parabolic systems including the compressible navier stokes and compressible mhd equations we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical navier stokes boundary conditions our first main result using the abstract framework established by the authors in the companion work gues et al in j differ equ 244 309 387 2008 is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an evans function uniform evans stability whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with residual boundary conditions our second is to show that uniform evans stability for small amplitude layers is equivalent to evans stability of the limiting constant layer which in turn can be checked by a linear algebraic computation finally for a class of symmetric dissipative systems including the physical examples mentioned above we carry out energy estimates showing that constant and thus small amplitude layers always satisfy uniform evans stability this yields existence of small amplitude multi dimensional boundary layers for the compressible navier stokes and mhd equations for both equations these appear to be the first such results in the compressible case

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

decay rates for the viscous incompressible mhd equations with and without surface tension abstract in this paper we consider a layer of a viscous incompressible electrically conducting fluid interacting with the magnetic field in a horizontally periodic setting the upper boundary is bounded by a free boundary and below bounded by a flat rigid interface we prove the global well posedness of the problem for both the case with and without surface tension moreover we show that the global solution decays to the equilibrium exponentially in the case with surface tension however the global solution decays to the equilibrium at an almost exponential rate in the case without surface tension

the euler limit of the navier stokes equations and rotating fluids with boundary in this paper we study the convergence of weak solutions of the navier stokes equations in some particular domains with different horizontal and vertical viscosities when they go to zero with different speeds the difficulty here comes from the dirichlet boundary conditions precisely we show that if the ratio of the vertical viscosity to the horizontal viscosity also goes to zero then the solutions converge to the solution of the euler system we study the same limit for rotating fluids with rossby number also going to zero

global well posedness of the free surface incompressible euler equations with damping abstract we consider a layer of an incompressible inviscid fluid bounded below by a fixed solid boundary and above by a free moving boundary in a horizontally periodic setting the fluid dynamics is governed by the gravity driven incompressible euler equations with damping and the effect of surface tension is neglected on the free surface we prove that the problem is globally well posed for the small initial data and that solutions decay to the equilibrium at an almost exponential rate

the zero surface tension limit of two dimensional water waves we consider two dimensional water waves of infinite depth periodic in the horizontal direction it has been proven by wu in the slightly different nonperiodic setting that solutions to this initial value problem exist in the absence of surface tension recently ambrose has proven that solutions exist when surface tension is taken into account in this paper we provide a shorter more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface using this formulation we find that as surface tension goes to 0 the water wave without surface tension is the limit of the water wave with surface tension far from being a simple adaptation of previous works our method requires a very original choice of variables these variables turn out to be physical and well adapted to both cases c

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

flops and clusters in the homological minimal model programme suppose that f x rightarrow mathrm spec r is a minimal model of a complete local gorenstein 3 fold where the fibres of f are at most one dimensional so by van den bergh duke math j 122 3 423 455 2004 there is a noncommutative ring lambda derived equivalent to x for any collection of curves above the origin we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of lambda is finite dimensional improving a result of donovan and wemyss contractions and deformations arxiv 1511 00406 we further show that the mutation functor of iyama and wemyss invent math 197 3 521 586 2014 6 is functorially isomorphic to the inverse of the bridgeland chen flop functor in the case when the factor of lambda is finite dimensional these results then allow us to jump between all the minimal models of mathrm spec r in an algorithmic way without having to compute the geometry at each stage we call this process the homological mmp this has several applications in git approaches to derived categories and also to birational geometry first using mutation we are able to compute the full git chamber structure by passing to surfaces we say precisely which chambers give the distinct minimal models and also say which walls give flops and which do not enabling us to prove the craw ishii conjecture in this setting second we are able to precisely count the number of minimal models and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity third we prove a bijective correspondence between maximal modifying r module generators and minimal models and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry this lifts the auslander mckay correspondence to dimension three

uniform regularity for the navier stokes equation with navier boundary condition we prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the navier stokes equation with the navier boundary condition has a strong solution this solution is uniformly bounded in a conormal sobolev space and has only one normal derivative bounded in l this allows us to obtain the vanishing viscosity limit to the incompressible euler system from a strong compactness argument

on the finiteness of the derived equivalence classes of some stable endomorphism rings we prove that the stable endomorphism rings of rigid objects in a suitable frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation the main application is to the homological minimal model programme for a 3 fold flopping contraction f x rightarrow mathrm spec r where x has only gorenstein terminal singularities there is an associated finite dimensional algebra a text con known as the contraction algebra as a corollary of our main result there are only finitely many basic algebras in the derived equivalence class of a text con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f this provides evidence towards a key conjecture in the area

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

uniform regularity and vanishing viscosity limit for the free surface navier stokes equations we study the inviscid limit of the free boundary navier stokes equations we prove the existence of solutions on a uniform time interval by using a suitable functional framework based on sobolev conormal spaces this allows us to use a strong compactness argument to justify the inviscid limit our approach does not rely on the justification of asymptotic expansions in particular we get a new existence result for the euler equations with free surface from the one for navier stokes

remarks about the inviscid limit of the navier stokes system in this paper we prove two results about the inviscid limit of the navier stokes system the first one concerns the convergence in hs of a sequence of solutions to the navier stokes system when the viscosity goes to zero and the initial data is in hs the second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions we present here a simple proof which also works in the 3d case the 3d case is new

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

on the ill posedness of the prandtl equation the concern of this paper is the cauchy problem for the prandtl equation this problem is known to be well posed for analytic data or for data with monotonicity properties we prove here that it is linearly ill posed in sobolev type spaces the key of the analysis is the construction at high tangential frequencies of unstable quasimodes for the linearization around solutions with non degenerate critical points interestingly the strong instability is due to vicosity which is coherent with well posedness results obtained for the inviscid version of the equation a numerical study of this instability is also provided

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

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

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

small time existence for the three dimensional navier stokes equations for an incompressible fluid with a free surface in 1980 beale 2 studied the problem of describing the motion of a layer of heavy viscous incompressible fluid lying above an infinite rigid bottom and having a non compact free surface with no surface tension he proved its local solvability in an anisotropic sobolev space for any initial data its global solvability was discussed by sylv sxer 20 but a crucial point of her proof does not appear clear to me the regularity of the free surface with surface tension taken into account beale 31 and beale nishida 4 respectively studied the large time existence and regularity and the large time behavior of the solution to this problem with initial data near equilibrium teramoto 24 25 considered these problems for fluids lying above an inclined plane for any initial data the same problem was solved locally in time by allain 11 in the two dimensional case the aim of this paper is to establish the analogous result in the three dimensional case a similar problem describing the motion of a finite isolated mass of incompressible viscous fluid was studied thoroughly by solonnikov he proved local solvability in a h61der space in 12 and global solvability in the space w 2 1 p n where n is the dimension of the domain in 16 without surface tension and in 13 15 17 19 21 with surface tension let us formulate our problem given an initial domain f r 3 with x3 being the vertical component and an initial velocity vector field vo in f we want to know the domain f t t 0 occupied by the fluid which is bounded by the fixed bottom sb and the free surface se t the velocity vector field v v x t vl v2 v3 and the pressure p p x t so that

on the free boundary problem of three dimensional incompressible euler equations in this paper we consider in three dimensions the motion of a general inviscid incompressible fluid with a free interface that separates the fluid region from the vacuum we assume that the fluid region is below the vacuum and that there is no surface tension on the free surface then we prove the local well posedness of the free boundary problem in sobolev space provided that there is no self intersection point on the initial surface and under the stability assumption that with being restricted to the initial surface 2007 wiley periodicals inc

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

reduction of triangulated categories and maximal modification algebras for ca_n singularities in this paper we define and study triangulated categories in which the homspaces have krull dimension at most one over some base ring hence they have a natural 2 step filtration and each factor of the filtration satisfies some calabi yau type property if c is such a category we say that c is calabi yau with dim c 1 we extend the notion of calabi yau reduction to this setting and prove general results which are an analogue of known results in cluster theory such categories appear naturally in the setting of gorenstein singularities in dimension three as the stable categories cm r of cohen macaulay modules we explain the connection between calabi yau reduction of cm r and both partial crepant resolutions and q factorial terminalizations of spec r and we show under quite general assumptions that calabi yau reductions exist in the remainder of the paper we focus on complete local can singularities r by using a purely algebraic argument based on calabi yau reduction of cm r we give a complete classification of maximal modifying modules in terms of the symmetric group generalizing and strengthening results in 7 10 where we do not need any restriction on the ground field we also describe the mutation of modifying modules at an arbitrary not necessarily indecomposable direct summand as a corollary when k d c we obtain many autoequivalences of the derived category of the q factorial terminalizations of spec r