analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

non uniform dependence for the periodic higher dimensional camassa holm equations abstract in this paper we investigate the dependence on initial data of solutions to the higher dimensional camassa holm equations with periodic boundary condition in besov spaces we show that when s 1 d 2 d 2 and 1 r the solution map is not uniformly continuous from b 2 r s t d into c 0 t b 2 r s t d for r or from b 2 s t d into l 0 t b 2 s t d for r

initial boundary value problems for nonlinear dispersive wave equations in this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half line and on a finite interval subject to homogeneous dirichlet boundary conditions we first prove local well posedness of the rod equation and of the b equation for general initial data furthermore we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span in the case of finite time singularities we are able to describe the precise blow up scenario of breaking waves our approach is based on sharp extension results for functions on the half line or on a finite interval and several symmetry preserving properties of the equations under discussion

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

crystal melting and toric calabi yau manifolds we construct a statistical model of crystal melting to count bps bound states of d0 and d2 branes on a single d6 brane wrapping an arbitrary toric calabi yau threefold the three dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of d branes the crystal is composed of atoms of different colors each of which corresponds to a node of the quiver diagram and the chemical bond is dictated by the arrows of the quiver diagram bps states are constructed by removing atoms from the crystal this generalizes the earlier results on the bps state counting to an arbitrary non compact toric calabi yau manifold we point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the donaldson thomas theory

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

non uniform continuous dependence on initial data of solutions to the euler poincare system in this paper we investigate the continuous dependence on initial data of solutions to the euler poincare system by constructing a sequence of approximate solutions and calculating the error terms we show that the data to solution map is not uniformly continuous in sobolev space hs rd for s 1 d2

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

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

wall crossing free fermions and crystal melting we describe wall crossing for local toric calabi yau manifolds without compact four cycles in terms of free fermions vertex operators and crystal melting firstly to each such manifold we associate two states in the free fermion hilbert space the overlap of these states reproduces the bps partition function corresponding to the non commutative donaldson thomas invariants given by the modulus square of the topological string partition function secondly we introduce the wall crossing operators which represent crossing the walls of marginal stability associated to changes of the b field through each two cycle in the manifold bps partition functions in non trivial chambers are given by the expectation values of these operators thirdly we discuss crystal interpretation of such correlators for this whole class of manifolds we describe evolution of these crystals upon a change of the moduli and find crystal interpretation of the flop transition and the dt pt transition the crystals which we find generalize and unify various other calabi yau crystal models which appeared in literature in recent years

wall crossing of bps states on the conifold from seiberg duality and pyramid partitions in this paper we study the relation between pyramid partitions with a general empty room configuration erc and the bps states of d branes on the resolved conifold we find that the generating function for pyramid partitions with a length n erc is exactly the same as the d6 d2 d0 bps partition function on the resolved conifold in particular kahler chambers we define a new type of pyramid partition with a finite erc that counts the bps degeneracies in certain other chambers

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

continuous properties of the solution map for the euler equations in this paper we study the dependence on initial data of solutions to the incompressible euler equations in besov spaces we show that for s n p 1 p 1 and r 1 the solution map u0 u is holder continuous in besov space bp rs equipped with weaker topology when the space variable x is taken to be periodic we obtain a family of explicit periodic solutions furthermore we prove that for any s r and 1 r the solution map is not globally uniformly continuous in b2 rs tn which extends some results of himonas and misiolek commun math phys 296 1 285 301 2010

on the noncommutative donaldson thomas invariants arising from brane tilings abstract given a brane tiling that is a bipartite graph on a torus we can associate with it a quiver potential and a quiver potential algebra under certain consistency conditions on a brane tiling we prove a formula for the donaldson thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra we relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling we prove that the same consistency conditions imply that the quiver potential algebra is a 3 calabi yau algebra we also formulate a rationality conjecture for the generating functions of the donaldson thomas type invariants

topological strings and integrable hierarchies we consider the topological b model on local calabi yau geometries we show how one can solve for the amplitudes by using w algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the calabi yau in the highly effective fermionic brane formulation this leads to a free fermion description of the amplitudes furthermore we argue that topological strings on calabi yau geometries provide a unifying picture connecting non critical super strings integrable hierarchies and various matrix models in particular we show how the ordinary matrix model the double scaling limit of matrix models and kontsevich like matrix model are all related and arise from studying branes in specific local calabi yau three folds we also show how a model topological string on p 1 and local toric threefolds and in particular the topological vertex can be realized and solved as b model topological string amplitudes on a calabi yau manifold

on the euler poincare equation with non zero dispersion we consider the euler poincare equation on mathbb r d d geqq 2 for a large class of smooth initial data we prove that the corresponding solution blows up in finite time this settles an open problem raised by chae and liu commun math phys 314 671 687 2012 our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the euler poincare flow in particular we show an abundance of blowups emanating from smooth initial data with certain sign properties no size restrictions are imposed on the data we also showcase a class of initial data for which the corresponding solution exists globally in time

blow up zero α limit and the liouville type theorem for the euler poincare equations in this paper we study the euler poincare equations in mathbb r n we prove local existence of weak solutions in w 2 p mathbb r n p n and local existence of unique classical solutions in h k mathbb r n k n 2 3 as well as a blow up criterion for the zero dispersion equation 0 we prove a finite time blow up of the classical solution we also prove that as the dispersion parameter vanishes the weak solution converges to a solution of the zero dispersion equation with sharp rate as 0 provided that the limiting solution belongs to c 0 t h k mathbb r n with k n 2 3 for the stationary weak solutions of the euler poincare equations we prove a liouville type theorem namely for 0 any weak solution mathbf u in h 1 mathbb r n is u 0 for 0 any weak solution mathbf u in l 2 mathbb r n is u 0