entropy solutions to a strongly degenerate anisotropic convection diffusion equation with application to utility theory we study the deterministic counterpart of a backward forward stochastic differential utility which has recently been characterized as the solution to the cauchy problem related to a pde of degenerate parabolic type with a conservative first order term we first establish a local existence result for strong solutions and a continuation principle and we produce a counterexample showing that in general strong solutions fail to be globally smooth afterward we deal with discontinuous entropy solutions and obtain the global well posedness of the cauchy problem in this class eventually we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data as a byproduct we establish the existence of an utility process which is a solution to a backward forward stochastic differential equation for a given class of final utilities which is relevant for financial applications

boundary layer for a class of nonlinear pipe flow we establish the mathematical validity of the prandtl boundary layer theory for a family of nonlinear parallel pipe flow the convergence is verified under various sobolev norms including the physically important space time uniform norm as well as thel 1 h 1 norm higher order asymptotics is also studied

the classification of affine su 3 modular invariant partition functions a complete classification of thephysical modular invariant partition functions for the wznw models is known for very few affine algebras and levels the most significant being all levels ofsu 2 and level 1 of all simple algebras in this paper we solve the classification problem forsu 3 modular invariant partition functions all levels our approach will also be applicable to other affine lie algebras and we include some preliminary work in that direction including a sketch of a new proof forsu 2

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

sublayer of prandtl boundary layers the aim of this paper is to investigate the stability of prandtl boundary layers in the vanishing viscosity limit nu to 0 in grenier commun pure appl math 53 9 1067 1091 2000 one of the authors proved that there exists no asymptotic expansion involving one of prandtl s boundary layer with thickness of order sqrt nu which describes the inviscid limit of navier stokes equations the instability gives rise to a viscous boundary sublayer whose thickness is of order nu 3 4 in this paper we point out how the stability of the classical prandtl s layer is linked to the stability of this sublayer in particular we prove that the two layers cannot both be nonlinearly stable in l infty that is either the prandtl s layer or the boundary sublayer is nonlinearly unstable in the sup norm

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

simple wzw currents a complete classification of simple currents of wzw theories is obtained the proof is based on an analysis of the quantum dimensions of the primary fields simple currents are precisely the primaries with unit quantum dimension for wzw theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible

extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls given a 1 periodic real potential q l1 r z we use 0 q to denote the smallest 1 periodic eigenvalue of the hill s equation x q t x 0 let b1 r be the ball centered at 0 of radius r in the l1 space l1 r z it is trivial that sup 0 q q b1 r r for all r 0 based on continuity of 0 q in q with the weak topology and continuous differentiability of 0 q in q with the l1 norm 1 we will apply scaling technique variational approach to extremal values in lp balls singular integrals and the limiting approach as p 1 to obtain i 0 q is bounded for q in any bounded set of l1 r z 1 and ii the minimal value l1 r inf 0 q q l1 r z q 1 r inf 0 q q l1 r z q 1 r is simply z0 1 r where z0 x 2 xtanh x 2 x 0 the extremal values of the smallest neumann eigenvalues for potentials in l1 balls are also found explicitly

continuous dependence and estimates of eigenvalues for periodic generalized camassa holm equations abstract in this paper we are concerned with the spectral problem y 1 y 2 x 1 2 1 2 m 1 2 m 1 2 y 1 y 2 for the periodic generalized camassa holm equations the first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials we prove that as nonlinear functionals of potentials eigenvalues are continuous in potentials with respect to the weak topologies in the lebesgue space l 1 0 t the second aim is to find the estimates of the eigenvalues when the l 1 norm of potentials is given based on an application of trace formulas

structural bifurcation of 2 d nondivergent flows with dirichlet boundary conditions applications to boundary layer separation this article addresses transitions in the topological structure of a family of divergence free vector fields u t with dirichlet boundary conditions we show that structural bifurcation i e change in topological equivalence class occurs at t0 if u t 0 has an isolated degenerate singular point x m such that 2 u x t0 n t 0 the main results are proved by classifying orbit structures of u near such a point x m of u t 0 the condition of x being a singular point is equivalent to the one originally postulated by prandtl for boundary layer separation our analysis and classification do contribute in fact to a rigorous characterization of boundary layer separation in 2 d incompressible fluid flows

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

dependence of solutions and eigenvalues of measure differential equations on measures abstract it is well known that solutions of ordinary differential equations are continuously dependent on finite dimensional parameters in equations in this paper we study the dependence of solutions and eigenvalues of second order linear measure differential equations on measures as an infinitely dimensional parameter we will provide two fundamental results which are the continuity and continuous frechet differentiability in measures when the weak topology and the norm topology of total variations for measures are considered respectively in some sense the continuity result obtained in this paper is the strongest one as an application we will give a natural simple explanation to extremal problems of eigenvalues of sturm liouville operators with integrable potentials

extremal values of eigenvalues of sturm liouville operators with potentials in l 1 balls abstract this paper is a continuation of zhang m zhang continuity in weak topology higher order linear systems of ode sci china ser a 51 2008 1036 1058 m zhang extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls j differential equations 246 2009 4188 4220 given a potential q l p 0 1 r p 1 we use m q to denote the m th dirichlet eigenvalue of the sturm liouville operator with potential q t where m n the minimal value l m p r and the maximal value m m p r of m q with potentials q in the l p ball of radius r are well defined in this paper we will exploit the continuity of m q in q with weak topologies and the variational method to give characterizations of l m p r and m m p r when p 1 by using the limiting approach as p 1 we find that the most important extremal values l m 1 r and m m 1 r can be evaluated explicitly using some elementary functions of r the corresponding extremal problems for neumann eigenvalues and some periodic eigenvalues will be reduced to l m p r and m m p r

local in time existence and uniqueness of solutions to the prandtl equations by energy methods we prove local existence and uniqueness for the two dimensional prandtl system in weighted sobolev spaces under oleinik s monotonicity assumption in particular we do not use the crocco transform or any change of variables our proof is based on a new nonlinear energy estimate for the prandtl system this new energy estimate is based on a cancellation property that is valid under the monotonicity assumption to construct the solution we use a regularization of the system that preserves this nonlinear structure this new nonlinear structure may give some insight into the convergence properties from the navier stokes system to the euler system when the viscosity goes to 0 2015 wiley periodicals inc

long time well posedness of prandtl system with small and analytic initial data abstract in this paper we investigate the long time existence and uniqueness of small solution to d for d 2 3 dimensional prandtl system with small initial data which is analytic in the horizontal variables in particular we prove that d dimensional prandtl system has a unique solution with the life span of which is greater than e 4 3 if the initial data is of size e and the value on the boundary of the tangential velocity of the outflow are of size e 5 3 we mention that the tool developed in 4 5 to make the analytical type estimates and the special structure of the nonlinear terms to this system play an essential role in the proof of this result

initial boundary layer associated with the nonlinear darcy brinkman oberbeck boussinesq system abstract in this paper we study the vanishing darcy number limit of the nonlinear darcy brinkman oberbeck boussinesq system dbob this singular perturbation problem involves singular structures both in time and in space giving rise to initial layers boundary layers and initial boundary layers we construct an approximate solution to the dbob system by the method of multiple scale expansions the convergence with optimal convergence rates in certain sobolev norms is established rigorously via the energy method

automorphism modular invariants of current algebras we consider those two dimensional rational conformal field theories rcfts whose chiral algebras when maximally extended are isomorphic to the current algebra formed from some untwisted affine lie algebra at fixed level in this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the kac peterson modular matrices we classify all such partition functions when the underlying finite dimensional lie algebra is simple this gives all possible spectra for this class of rcfts while accomplishing this we also find the primary fields with second smallest quantum dimension

initial boundary layer associated with the nonlinear darcy brinkman system abstract we study the interaction of initial layer and boundary layer in the nonlinear darcy brinkman system in the vanishing darcy number limit in particular we show the existence of a function of corner layer type so called initial boundary layer in the solution of the nonlinear darcy brinkman system an approximate solution is constructed by the method of multiple scale expansion in space and in time we establish the optimal convergence rates in various sobolev norms via energy method

symmetrical prandtl boundary layer expansions of steady navier stokes equations on bounded domain abstract this paper is concerned with the validity of the prandtl boundary layer theory in the inviscid limit of the steady incompressible navier stokes equations which is an extension of the pioneer paper 13 y guo et al 2017 ann pde from a domain of 0 l r to 0 l 0 2 under the symmetry assumption we establish the validity of the prandtl boundary layer expansions and the error estimates the convergence rate as e 0 is also given

a well posedness theory for the prandtl equations in three space variables abstract the well posedness of the three space dimensional prandtl equations is studied under some constraint on its flow structure together with the instability result given in 28 it gives an almost necessary and sufficient structural condition for the stability of the three dimensional prandtl equations it reveals that the classical burgers equation plays an important role in determining this type of flow with special structure that avoids the appearance of the secondary flow an unstabilizing factor in the three dimensional prandtl boundary layers and the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the prandtl boundary layer equations is illustrated in the three dimensional setting moreover it is shown that this structured flow is linearly stable for any smooth three dimensional perturbation

on the spectral problem for the periodic camassa holm equation w x was recently derived 2 to describe the motion of solitary waves on shallow water it presents interesting features that do not arise in the well known korteweg de vries kdv model for example the solitary wave for the camassa holm equation has a corner at its peak the interaction of two w x solitary waves also presents striking differences with the case of kdv 2 let us consider the associated spectral problem in which the time t of the camassa holm equation appears only as a parameter

well posedness of the prandtl equation in sobolev spaces we develop a new approach to study the well posedness theory of the prandtl equation in sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the crocco transformation precisely we firstly investigate the linearized prandtl equation in some weighted sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation we apply the nash moser hormander iteration to obtain a well posedness theory of classical solutions to the nonlinear prandtl equation when the initial data is a small perturbation of a monotonic shear flow

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

z nz conformal field theories in this paper we give a brief introduction to conformal field theory cft following the presentation of g segal we explain how to reconstruct part of a cft from its fusion rules the possible choices of s matrices are indexed by some automorphisms of the fusion algebra we illustrate this procedure by computing the modular properties of the possible genus one characters when the fusion algebra is the representation algebra of a finite group we also classify the modular invariant partition functions of these theories we recover as special cases the a n 1 wzw theories and the rational gaussian model

almost global existence for the prandtl boundary layer equations we consider the prandtl boundary layer equations on the half plane with initial datum that lies in a weighted h1 space with respect to the normal variable and is real analytic with respect to the tangential variable the boundary trace of the horizontal euler flow is taken to be a constant we prove that if the prandtl datum lies within varepsilon of a stable profile then the unique solution of the cauchy problem can be extended at least up to time t varepsilon geqq rm exp varepsilon 1 rm log varepsilon 1

breakdown of the camassa holm equation the camassa holm equation is written in its eulerian form as displaystyle begin array rcl text ch quad frac partial v partial t v frac partial v partial x frac partial p partial x 0 end array with pressure p g v 2 frac 1 2 v 2 here g 1 d 2 1 frac 1 2 e vert x y vert it is easy to see that if at time t 0 v is odd with v x 0 for x 0 and v 0 0 then the slope s t v t 0 satisfies s bullet frac 1 2 s 2 and so is driven down to infty at some time t leq 2 s 0 in short the flow may break down this is the steepening lemma of camassa and holm 2 nothing worse happens v t x itself cannot jump like the usual kind of shock actually it is not the obvious v but rather the auxiliary function m v v that controls this m is a caricature of the vorticity that controls the behavior of honest euler in dimensions 2 and 3 see bertozzi and majda 1 for a full account

long time well posdness of the prandtl equations in sobolev space in this paper we study the long time well posedness for the nonlinear prandtl boundary layer equation on the half plane while the initial data are small perturbations of some monotonic shear profile we prove the existence uniqueness and stability of solutions in weighted sobolev space by energy methods the key point is that the life span of the solution could be any large t as long as its initial date is a perturbation around the monotonic shear profile of small size like e t the nonlinear cancellation properties of prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate

boundary layer associated with the darcy brinkman boussinesq model for convection in porous media abstract we study the asymptotic behavior of the infinite darcy prandtl number darcy brinkman boussinesq system for convection in porous media at a small brinkman darcy number the existence of a boundary layer with thickness proportional to the square root of the brinkman darcy number for the velocity field is established in both the l h 1 norm in 2 and 3 d and the l l norm in 2d this improves in several respects an earlier result of payne and straughan 1998 60 where the vanishing brinkman darcy number limit is studied without resolving the boundary layer

zero viscosity limit of the linearized navier stokes equations for a compressible viscous fluid in the half plane the zero viscosity limit for an initial boundary value problem of the linearized navier stokes equations of a compressible viscous fluid in the half plane is studied by means of the asymptotic analysis with multiple scales we first construct an approximate solution of the linearized problem of the navier stokes equations as the combination of inner and boundary expansions next by carefully using the technique on energy methods we show the pointwise estimates of the error term of the approximate solution which readily yield the uniform stability result for the linearized navier stokes solution in the zero viscosity limit 1999 john wiley sons inc

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

vanishing viscosity limit for incompressible navier stokes equations with navier boundary conditions for small slip length in this paper we will analyze the vanishing viscosity limit of the incompressible navier stokes equations with the navier slip boundary conditions in a bounded domain of r2 when the slip length is smaller than or equal to the order of viscosity by using an energy method and developing kato s approach given in the work of kato math sci res inst publ 2 85 98 1984 we obtain several conditions to guarantee that the solution of the navier stokes equations with the navier slip boundary conditions goes to the solution of the associated problem of the euler equations in the energy space l2 uniformly in time as the viscosity goes to zero

zero viscosity limit for analytic solutions of the primitive equations the aim of this paper is to prove that the solutions of the primitive equations converge in the zero viscosity limit to the solutions of the hydrostatic euler equations we construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic euler equation and boundary layer correctors as the first order term and an error that we show to be o sqrt nu the main assumption is spatial analyticity of the initial datum

discrete kato type theorem on inviscid limit of navier stokes flows the inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics here we present a somewhat surprising result related to numerical approximation of the problem more precisely we show that numerical solutions of the incompressible navier stokes equations converge to the exact solution of the euler equations at vanishing viscosity and vanishing mesh size provided that small scales of the order of u in the directions tangential to the boundary are not resolved in the scheme here is the kinematic viscosity of the fluid and u is the typical velocity taken to be the maximum of the shear velocity at the boundary for the inviscid flow such a result is somewhat counterintuitive since the convergence is ensured even in the case that small scales predicted by the conventional theory of turbulence and boundary layer are not resolved since under resolution which is allowed in our theorem in advection dominated problem usually leads to oscillation whic

finite time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics in an earlier work we have shown the global for all initial data and all time well posedness of strong solutions to the three dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics in this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid non viscous primitive equations if they exist they blow up in finite time specifically we consider the three dimensional inviscid primitive equations in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no normal flow boundary conditions on the solid top and bottom boundaries for certain class of initial data we reduce this system into the two dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions and then we show that for specific sub class of initial data the corresponding smooth solutions of the reduced inviscid two dimensional system develop singularities in finite time

steady prandtl boundary layer expansions over a rotating disk this paper concerns the validity of the prandtl boundary layer theory for steady incompressible navier stokes flows over a rotating disk we prove that the navier stokes flows can be decomposed into euler and prandtl flows in the inviscid limit in so doing we develop a new set of function spaces and prove several embedding theorems which capture the interaction between the prandtl scaling and the geometry of our domain

on the ill posedness of the prandtl equations in three dimensional space in this paper we give an instability criterion for the prandtl equations in three dimensional space which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well posedness of the three dimensional prandtl equations in contrast to the classical well posedness theory of the two dimensional prandtl equations under the oleinik monotonicity assumption both linear stability and nonlinear stability are considered this criterion shows that the monotonic shear flow is linearly stable for the three dimensional prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable and this result is an exact complement to our recent work a well posedness theory for the prandtl equations in three space variables arxiv 1405 5308 2014 on the well posedness theory for the three dimensional prandtl equations with a special structure

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

on the isospectral problem of the dispersionless camassa holm equation abstract we discuss direct and inverse spectral theory for the isospectral problem of the dispersionless camassa holm equation where the weight is allowed to be a finite signed measure in particular we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite the results are applied to deduce several facts for the dispersionless camassa holm equation in particular we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by mckean

the van dommelen and shen singularity in the prandtl equations abstract in 1980 van dommelen and shen provided a numerical simulation that predicted the spontaneous generation of a singularity in the prandtl boundary layer equations from a smooth initial datum for a nontrivial euler background in this paper we prove this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness

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

a blow up criterion for classical solutions to the prandtl equations in this paper we obtain a blow up criterion for classical solutions to the two dimensional prandtl equations under a monotonicity assumption in weighted sobolev spaces our proof is based on nonlinear energy estimates inspired by masmoudi and wong commun pure appl math 68 1683 1741 2015 and maximum principles for parabolic equations

magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces abstract in this paper we are concerned with the magnetic effect on the sobolev solvability of boundary layer equations for the 2d incompressible mhd system without resistivity the mhd boundary layer is described by the prandtl type equations derived from the incompressible viscous mhd system without resistivity under the no slip boundary condition on the velocity assuming that the initial tangential magnetic field does not degenerate a local in time well posedness in sobolev spaces is proved without the monotonicity condition on the velocity field moreover we show that if the tangential magnetic field of shear layer is degenerate at one point then the linearized mhd boundary layer system around the shear layer profile is ill posed in the gevrey function space provided that the initial velocity shear flow is non degenerately critical at the same point

local in time well posedness for compressible mhd boundary layer abstract in this paper we are concerned with the motion of electrically conducting fluid governed by the two dimensional non isentropic viscous compressible mhd system on the half plane with no slip condition on the velocity field perfectly conducting wall condition on the magnetic field and dirichlet boundary condition on the temperature on the boundary when the viscosity heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate there is a boundary layer which is described by a prandtl type system under the non degeneracy condition on the tangential magnetic field instead of monotonicity of velocity by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work 27 we obtain the local in time well posedness of the boundary layer system in weighted sobolev spaces

on the global existence of solutions to the prandtl s system abstract in this paper we establish a global existence of weak solutions to the two dimensional prandtl s system for unsteady boundary layers in the class considered by oleinik j appl math mech 30 1966 951 provided that the pressure is favourable this generalizes the local well posedness results due to oleinik 1966 mathematical models in boundary layer theory chapman hall london 1999 for the proof we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations

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

continuity in weak topology and extremal problems of eigenvalues of the p laplacian we will study the dependence of eigenvalues of the one dimensional p laplacian on potentials or weights two results are obtained one is the continuity of eigenvalues in potentials with respect to the weak topologies of l spaces 1 and the other is the continuous differentiability of eigenvalues in potentials with respect to l norms as applications we will study some extremal problems of eigenvalues by developing some analytical methods

boundary layer separation and adverse pressure gradient for 2 d viscous incompressible flow we study the detailed process of bifurcation in the flow s topological structure for a two dimensional 2 d incompressible flow subject to no slip boundary conditions and its connection with boundary layer separation the boundary layer separation theory of m ghil t ma and s wang based on the structural bifurcation concept is translated into vorticity form the vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary this singular point is characterized by nonzero tangential second order derivative and nonzero time derivative of the vorticity furthermore we prove the presence of an adverse pressure gradient at the critical point due to reversal in the direction of the pressure force with respect to the basic shear flow at this point a numerical example of 2 d driven cavity flow governed by the navier stokes equations is presented boundary layer separation occurs the bifurcation criterion is satisfied and an adverse pressure gradient is shown to be present

local existence and uniqueness of euler equation in weighted sobolev space abstract in this paper we study the existence and uniqueness of euler equation in the weighted sobolev space this is motivated by the study of inviscid limit of navier stokes equation in half plane the use of the weighted calderon zygmund inequality in the estimate of the pressure term is the improvement of our work

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

from the boltzmann equation to the euler equations in the presence of boundaries the fluid dynamic limit of the boltzmann equation leading to the euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the navier stokes equations the present paper slightly extends recent results from c bardos f golse l paillard the incompressible euler limit of the boltzmann equation with accommodation boundary condition comm math sci 10 2012 159 190 to the case of boundary conditions for the boltzmann equation more general than maxwell s accommodation condition

automorphisms of the affinesu 3 fusion rules we classify the automorphisms of the chiral level k affinesu 3 fusion rules for any value ofk by looking for all permutations that commute with the modular matricess andt this can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed whenk is divisible by 3 the automorphism group z2 is generated by the charge conjugationc ifk is not divisible by 3 the automorphism group z2 z2 is generated byc and the altschuler lacki zaugg automorphism although the combinatorial analysis can become more involved the techniques used here forsu 3 can be applied to other algebras

ill posedness of the hydrostatic euler and navier stokes equations we prove that the linearization of the hydrostatic euler equations at certain parallel shear flows is ill posed the result also extends to the hydrostatic navier stokes equations with a small viscosity

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation