a remark on regularity criterion for the dissipative quasi geostrophic equations abstract this paper concerns with a regularity criterion of solutions to the 2d dissipative quasi geostrophic equations based on a logarithmic sobolev inequality in besov spaces the absence of singularities of in 0 t is derived for a solution on the interval 0 t satisfying the condition l r 0 t b p 0 for 2 p r 4 p this is an extension of earlier regularity results in the serrin s type space l r 0 t l p

global solutions to the two dimensional quasi geostrophic equation with critical or super critical dissipation the two dimensional quasi geostrophic 2d qg equation with critical and super critical dissipation is studied in sobolev space h s 2 for critical case open image in new window existence of global large solutions in h s is proved for s open image in new window when open image in new window is small this generalizes and improves the results of constantin d cordoba and wu 4 for s 1 2 and the result of a cordoba and d cordoba 8 for s open image in new window for s 1 these solutions are also unique the improvement for pushing s down from 1 to open image in new window is somewhat surprising and unexpected for super critical case 0 open image in new window existence and uniqueness of global large solution in h s is proved when the product open image in new window is small for suitable s 2 2 p 1 and 0 1

generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

well posedness and decay of solutions for three dimensional generalized navier stokes equations abstract in this paper we study the generalized incompressible navier stokes equations in r 3 based on the energy estimates and regularization of the initial data with the heat semigroup we prove the well posedness of solutions in h 5 4 2 r 3 provided that the h 5 4 2 r 3 norm of initial data is sufficiently small in addition in contrast to the generalized heat equation the upper bound of the time decay rate of solutions to the generalized navier stokes equations is also established

on global regularity of 2d generalized magnetohydrodynamic equations abstract in this article we study the global regularity of 2d generalized magnetohydrodynamic equations 2d gmhd in which the dissipation terms are u and b we show that smooth solutions are global in the following three cases 1 2 1 0 1 2 2 2 2 0 we also show that in the inviscid case 0 if 1 then smooth solutions are global as long as the direction of the magnetic field remains smooth enough

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

on uniqueness for a semilinear parabolic system coupled in an equation and a boundary condition abstract we consider the system u t u v p v t v x r n t 0 u x 1 0 v x 1 u q x 1 0 t 0 u x 0 u 0 x v x 0 v 0 x x r n where r n x 1 x x r n 1 x 1 0 p q are positive numbers and functions u0 v0 in the initial conditions are nonnegative and bounded we show that nonnegative solutions are unique if pq 1 we also find a nontrivial nonnegative solution with vanishing initial values when pq

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

dissipative quasi geostrophic equation for large initial data in the critical sobolev space the critical and super critical dissipative quasi geostrophic equations are investigated in mathbb r 2 we prove local existence of a unique regular solution for arbitrary initial data in h2 2 which corresponds to the scaling invariant space of the equation we also consider the behavior of the solution near t 0 in the sobolev space

asymptotic convergence rate of supercritical surface quasi geostrophic equation in morrey space abstract in this paper the asymptotic convergence rate of the supercritical surface quasi geostrophic equation in morrey space is investigated it is shown that if the global solution x t of the supercritical surface quasi geostrophic equation satisfies the growth condition in critical morrey space 0 t m p q r 2 r d t for r 2 p 2 p then every perturbed weak solution x t of the perturbed quasi geostrophic equation converges asymptotically to x t with the optimal rate t t l 2 o 1 t 1 as t in particular the initial perturbation is allowed large the findings are based on the spectral decomposition methods of fractional laplace and morrey space technique

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

a necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations in their 1968 paper fujita and watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded non negative solutions in particular they showed that if the underlying ode has non unique solutions as characterised via an osgood type condition and the nonlinearity f satisfies a concavity condition then the parabolic pde also inherits the non uniqueness property this concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other similar non uniqueness phenomena in parabolic equations in this paper we provide an elementary proof of non uniqueness for the pde without any such concavity assumption on f an important consequence of our result is that uniqueness of the trivial solution of the pde is equivalent to uniqueness of the trivial solution of the corresponding ode which in turn is known to be equivalent to an osgood type integral condition on f

on the blow up criterion for the quasi geostrophic equations in homogeneous besov spaces abstract in this paper we consider the blow up criterion for the quasi geostrophic equations with dissipation 0 1 by establishing a new trilinear estimate we show that if l s 1 0 t b s r 2 for some s 1 2 1 then the solution can be extended smoothly past t this improves and extends the corresponding results in dong and pavlovic 2009 32 and yuan 2010

regularity criteria for the dissipative quasi geostrophic equations in holder spaces we study regularity criteria for weak solutions of the dissipative quasi geostrophic equation with dissipation 2 0 1 we show in this paper that if theta in c 0 t c 1 gamma or theta in l r 0 t c alpha with alpha 1 gamma frac gamma r is a weak solution of the 2d quasi geostrophic equation then is a classical solution in 0 t times mathbb r 2 this result improves our previous result in 18

global well posedness for the 2d inviscid benard system with fractional diffusivity and yudovichʼs type data abstract in this paper we consider the cauchy problem of the two dimensional inviscid benard system with fractional diffusivity we show that there is a global unique solution to this system with yudovich s type data

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

existence and regularity of weak solutions to the quasi geostrophic equations in the spaces lp or dot h 1 2 in this paper we study the 2d quasi geostrophic equation with and without dissipation we give global existence results of weak solutions for an initial data in the space lp or dot h 1 2 in the dissipative case when the initial data is in lp p 2 we give a regularity result of these solutions

symmetry of components and liouville type theorems for semilinear elliptic systems involving the fractional laplacian abstract in this paper we study the symmetry of components of positive solutions for the fractional elliptic systems in the whole space r n as an application we prove the uniqueness of positive solutions for fractional schrodinger type systems with critical exponents we also prove the symmetry of components and some liouville type theorems for a class of fractional noncooperative systems which contains the systems arising in bose einstein condensation and in biological models of lotka volterra type

on the a priori estimates for the euler the navier stokes and the quasi geostrophic equations abstract we prove new a priori estimates for the 3d euler the 3d navier stokes and the 2d quasi geostrophic equations by the method of similarity transforms

existence of solutions for an inhomogeneous fractional semilinear heat equation abstract we obtain necessary conditions and sufficient conditions on the existence of solutions to the cauchy problem for a fractional semilinear heat equation with an inhomogeneous term we identify the strongest spatial singularity of the inhomogeneous term for the solvability of the cauchy problem

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

on the well posedness of various one dimensional model equations for fluid motion abstract we consider 1d equations with nonlocal velocity of the form w t u w x u x w w where the nonlocal velocity u is given by 1 u 1 x x w 0 or 2 u h w h is the hilbert transform in this paper we address several local well posedness results with blow up criteria for smooth initial data we then establish the global well posedness by using the blow up criteria

the 2d regularized incompressible boussinesq equations with general critical dissipations abstract considering the 2d regularized boussinesq equations with fractional dissipations u and convection terms u u u where and 0 we prove the global existence and uniqueness of the solution in two critical cases the first case has fractional dissipations u where 1 0 and the second one has particular dissipation 1 u 0 in particular for the case 0 we give some decay estimates for u and the uniform estimate for g independent of time where g 1 u 2 2 u 1 1

commutator estimate in terms of partial derivatives of solutions for the dissipative quasi geostrophic equation through littlewood paley decomposition argument a commutator estimate in terms of partial derivatives of solutions for the critical and supercritical dissipative 2d quasi geostrophic equation is established as an application of this estimate we obtain some new a priori estimates and prove the existence and uniqueness of solution for the small initial data in critical besov spaces

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

nonexistence of simple hyperbolic blow up for the quasi geostrophic equation the problem we are concerned with is whether singularities form in finite time in incompressible fluid flows it is well known that the answer is no in the case of euler and navier stokes equations in dimension two in dimension three it is still an open problem for these equations in this paper we focus on a two dimensional active scalar model for the 3d euler vorticity equation constantin majda and tabak 7 suggested by studying rigorous theorems and detailed numerical experiments a general principle if the level set topology in the temperature field for the 2d quasi geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle then no finite time singularity is possible numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle there is a naturally associated notion of simple hyperbolic saddle breakdown the main theorem we present in this paper shows that such breakdown cannot occur in finite time we also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time using the same techniques we see that analogous results hold for incompressible 2d and 3d euler these results were announced in 9 but with a slight difference in the definition of a simple hyperbolic saddle the definition given in section 4 generalizes the one given in the announcement see also constantin 4 discussed in section 7 remark 5 below the whole work described in this paper is basically part of the author s thesis i am particularly grateful to my thesis advisor charles fefferman for his attention support guidance and advice i am indebted to d christodoulou and p constantin for helpful corrections and suggestions i wish to thank a majda for suggesting the subject and e tabak for discussions and com

on regularity criterion for the dissipative quasi geostrophic equations abstract in this paper we consider the 2d dissipative quasi geostrophic equations and study the regularity criterion of the solutions by means of a commutator estimate based on frequency localization and bony s paraproduct decomposition we obtain a regularity criterion l r 0 t b p 0 for 2 p r 2 p which is an obvious extension of the result in b dong z chen a remark on regularity criterion for the dissipative quasi geostrophic equations j math anal appl 329 2007 1212 1217 where their restrictive condition on p is 4 p

remarks on the global regularity for the super critical 2d dissipative quasi geostrophic equation abstract in this article we apply the method used in the recent elegant proof by kiselev nazarov and volberg of the well posedness of critically dissipative 2d quasi geostrophic equation to the super critical case we prove that if the initial value satisfies 0 l 1 2 s 0 l 2 s c s for some small number c s 0 where s is the power of the fractional laplacian then no finite time singularity will occur for the super critically dissipative 2d quasi geostrophic equation

syzygetic ideals regular sequences and a question of simis let a be a noetherian ring and i an ideal of a simis 7 posed the following question if torf z a z is an a z free module what consequences are brought on i in the present article we consider this question in local noetherian rings our purpose is to give some reasonable additional conditions which ensure that z is generated by a regular sequence the paper is divided into two parts the first one is devoted to obtaining the following result

geometric constrains for global regularity of 2d quasi geostrophic flows we study the two dimensional quasi geostrophic equations 2d qg in sobolev spaces we first prove a new analytic condition for global regularity which is both sufficient and necessary we then prove several new results on the geometric constraints on the 2d qg active scalar which suppress the development of singularity from the nonlinear stretching mechanism we focus mainly on the case with critical dissipation our results are also relevant to the inviscid case

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

global regularity of the two dimensional incompressible generalized magnetohydrodynamics system abstract in this paper we consider the global regularity of the two dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent 1 4 and magnetic diffusion exponent 1 based on the improved integrability in terms of vorticity this result is a further improvement of very recent works jiu and zhao 2014 and yamazaki 2013 which require 1 3 and 1 therefore our result is more closer to the resolution of the well known global regularity issue on the 2d magnetohydrodynamics system 0 1

long time dynamics of 2d quasi geostrophic equations with damping in lp abstract the long time behavior of solutions of 2d quasi geostrophic equations with damping on the whole space is studied the existence of the global attractor in l p is proved new difficulties come from nonlocal terms and the banach space l p we overcome them by using commutator estimates and positivity inequalities associated with fractional laplacian

dynamics of a fractional hydrodynamical equation for the heisenberg paramagnet in the present work we study the global solvability and large time dynamics for a fractional generalization of the hydrodynamical equation modeling the soft micromagnetic materials introducing a cancellation property we prove the existence of weak solutions and establish a uniqueness criterion a maximal principle is obtained and the global existence and uniqueness of smooth solutions are proved by some a priori estimates finally we analyze the asymptotic behavior of the solutions within the theory of infinite dimensional dissipative dynamical systems we prove that the problem generates a strongly continuous semigroup on a suitable phase space and show the existence of a maximal global attractor a in this phase space moreover in absence of external force global attractor a converges exponentially to a single equilibrium

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

a two dimensional model for quasigeostrophic flow comparison with the two dimensional euler flow abstract a simple two dimensional model for quasigeostrophic flow is contrasted with the two dimensional incompressible euler equations the model arises under the assumptions of fast rotation uniform stratification and uniform potential vorticity it is found that the more loca feed back of the quasigeostrophic model gives rise to strongly nonlinear front formation as opposed to two dimensional euler where the steepening process of mature fronts obeys a nonlocal nearly linear mechanism

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

existence and uniqueness of the solution to the dissipative 2d quasi geostrophic equations in the sobolev space we study the two dimensional dissipative quasi geostrophic equations in the sobolev space open image in new window existence and uniqueness of the solution local in time is proved in h s when s 2 1 existence and uniqueness of the solution global in time is also proved in h s when s 2 1 and the initial data open image in new window is small for the case s 2 1 we also obtain the unique large global solution in h s provided that open image in new window is small enough

gevrey regularity for the supercritical quasi geostrophic equation abstract in this paper following the techniques of foias and temam we establish suitable gevrey class regularity of solutions to the supercritical quasi geostrophic equations in the whole space with initial data in critical sobolev spaces moreover the gevrey class that we obtain is near optimal and as a corollary we obtain temporal decay rates of higher order sobolev norms of the solutions unlike the navier stokes or the subcritical quasi geostrophic equations the low dissipation poses a difficulty in establishing gevrey regularity a new commutator estimate in gevrey classes involving the dyadic littlewood paley operators is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty

a remark on global regularity of 2d generalized magnetohydrodynamic equations abstract in this paper we study the global regularity of the following 2d two dimensional generalized magnetohydrodynamic equations u t u u p b b u b t u b b u b and get global regular solutions when 0 1 2 1 3 2 3 which improves the results in 5 in particular we obtain the global regularity of the 2d generalized mhd when 0 and 3 2

a logarithmically improved regularity criterion for the surface quasi geostrophic equation abstract in this paper we consider the two dimensional dissipative surface quasi geostrophic equation and establish a logarithmically improved regularity criterion consequently our result extends the regularity criterion result of dong and pavlovie 2009 in addition a logarithmically improved regularity criterion to the three dimensional navier stokes equations is also derived by the same arguments therefore this result extends and improves many previous works

l2 stability for solutions of subcritical quasi geostrophic equation with large perturbations abstract this paper concerns with the asymptotic stability for the weak solution of the two dimensional dissipative subcritical quasi geostrophic equation in the class l p 0 t l q r 2 for p 2 q 2 q 1 2 it is proved that although the initial data and external force perturbations from are large every perturbed weak solution converges asymptotically to in l 2 r 2 as t as a byproduct we also obtain the weak strong uniqueness criterion of subcritical quasi geostrophic equation which improves the earlier results

a uniqueness result for a semilinear parabolic system abstract we prove that for nonnegative continuous bounded and nonzero initial data we have a unique solution of the reaction diffusion system described by three differential equations with non lipschitz nonlinearity we also find the set of all nonnegative solutions of the system when the initial data is zero and in the last section we briefly discuss a generalization of the theorem to a system of n equations

a pointwise estimate for fractionary derivatives with applications to partial differential equations this article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and lp decay of solutions of several interesting partial differential equations in particular there are applications to quasigeostrophic flows in two space variables with critical viscosity that model the eckman pumping see baroud ch n plapp b b she z s swinney h l 2002 phys rev lett 88 114501 and constantin p 2002 phys rev lett 89 184501

nonexistence of global solutions for a class of two time nonlinear evolution equations nonexistence results of global solutions for a class of ultraparabolic equations and systems involving the fractional laplacian operator are derived the proofs are based on the choice of a suitable test function in the weak formulation of the considered problems the considered equations and systems have their applications in diffusion theory in porous media

the maximum principle and the global attractor for the dissipative 2d quasi geostrophic equations the long time behavior of the solutions to the two dimensional dissipative quasi geostrophic equations is studied we obtain a new positivity lemma which improves a previous version of a cordoba and d cordoba 10 and 11 as an application of the new positivity lemma we obtain the new maximum principle i e the decay of the solution in l p for any p 2 when f is zero as a second application of the new positivity lemma for the sub critical dissipative case with open image in new window the existence of the global attractor for the solutions in the space h s for any s 2 1 is proved for the case when the time independent f is non zero therefore the global attractor is infinitely smooth if f is this significantly improves the previous result of berselli 2 which proves the existence of an attractor in some weak sense for the case 1 the global attractor exists in h s for any s 0 and the estimate of the hausdorff and fractal dimensions of the global attractor is also available

boundedness and blow up for a semilinear reaction diffusion system abstract we consider the semilinear parabolic system s u t u p t u q where x rnn 1 t 0 and p q are positive real numbers at t 0 nonnegative continuous and bounded initial values u0 v0 x are prescribed the corresponding cauchy problem then has a nonnegative classical and bounded solution u t x v t x in some strip st 0 t r n 0 t sup t 0 u v remain bounded in s t we show in this paper that if 0 t so that solutions can be continued for all positive times when pq 1 and 1 pq 1 n 2 with max p q one has t for every nontrivial solution u v t is then called the blow up time of the solution under consideration finally if 1 pq 1 n 2 both situations coexist since some nontrivial solutions remain bounded in any strip s while others exhibit finite blow up times

on the regularity criteria of a surface quasi geostrophic equation abstract we obtain regularity criteria for a quasi geostrophic equation that depends more on one direction than the others in particular we show that in the critical case the global regularity depends only on a partial derivative rather than a gradient of the solution

existence and uniqueness results for the 2 d dissipative quasi geostrophic equation abstract this paper concerns itself with besov space solutions of the 2 d quasi geostrophic qg equation with dissipation induced by a fractional laplacian the goal is threefold first to extend a previous result on solutions in the inhomogeneous besov space b 2 q r j wu global solutions of the 2d dissipative quasi geostrophic equation in besov spaces siam j math anal 36 2004 2005 1014 1030 to cover the case when r 2 2 second to establish the global existence of solutions in the homogeneous besov space b p q r with general indices p and q and third to determine the uniqueness of solutions in any one of the four spaces b 2 q s b p q r l q 0 t b 2 q s 2 q and l q 0 t b p q r 2 q where s 2 2 and r 1 2 2 p

upper and lower bounds of decay rates for h 12 weak solutions of the surface quasi geostrophic equation abstract this study is concerned with the large time decay estimates for weak solutions of the surface quasi geostrophic equation with h 1 2 initial data due to some special observation together with applying the decay characters to the class of h 1 2 functions we may show the upper and lower bounds of decay rates for the h 1 2 weak solution of the surface quasi strophic equation moreover as a 1 2 order derivative of h 1 2 weak solution we further derive an improved upper and lower bounds of decay rates for l 2 weak solution the findings may be of independent interest and the technique can be also applied to the other related fluid dynamical model with solutions in the negative sobolev spaces

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

asymptotic behavior for the quasi geostrophic equations with fractional dissipation in r2 abstract in this paper the surface quasi geostrophic equations qge with fractional dissipation in r 2 are considered our aim is to study the long time behavior of solutions of qge in the subcritical case to this end we investigate the global well posedness and global attractor for qge in h s r 2 via commutator estimates for nonlinear terms a new iterative technique for estimates of higher order derivatives and with the help of a nonlocal damping term besides by using the fractional lieb thirring inequality estimates of the finite hausdorff and fractal dimensions of the global attractor are found

lyapunov functions identities and the cauchy problem for the hele shaw equation this article is devoted to the study of the hele shaw equation we introduce an approach inspired by the water wave theory starting from a reduction to the boundary introducing the dirichlet to neumann operator and exploiting various cancellations we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface this allows to quasi linearize the equations in a natural way by combining these exact identities with convexity inequalities we prove the existence of hidden lyapunov functions of different natures we also deduce from these identities and previous works on the water wave problem a simple proof of the well posedness of the cauchy problem the analysis contains two side results of independent interest firstly we give a principle to derive estimates for the modulus of continuity of a pde under general assumptions on the flow secondly we prove and give applications of a convexity inequality for the dirichlet to neumann operator

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

growth of solutions for qg and 2d euler equations the work of constantin majda tabak 1 developed an analogy between the quasi geostrophic and 3d euler equations constantin majda and tabak proposed a candidate for a singularity for the quasi geostrophic equation their numerics showed evidence of a blow up for a particular initial data where the level sets of the temperature contain a hyperbolic saddle the arms of the saddle tend to close in finite time producing a sharp front numerics studies done later by ohkitani yamada 8 and constantin nie schorghofer 2 with the same initial data suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time the study of collapse on a curve was first studied in 1 for the quasi geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology at the time of collapse the scalar is discontinuous across the curve x2 f x1 with different limiting values for the temperature on each side of the front they show that under this topology the directional field remains smooth up to the collapse which contradicts the following theorem proven in 1