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

the global existence and time decay for the solutions of the fractional pseudo parabolic equation abstract we consider the cauchy problem of fractional pseudo parabolic equation on the whole space r n n 1 here the fractional order is related to the diffusion type source term behaving as the usual diffusion term on the high frequency part it has a feature of regularity gain and regularity loss for 1 and 0 1 respectively we establish the global existence and time decay rates for small amplitude solutions to the cauchy problem for 0 in the case that 0 1 we introduce the time weighted energy method to overcome the weakly dissipative property of the equation

global regularity for the supercritical dissipative quasi geostrophic equation with large dispersive forcing we consider the 2d quasi geostrophic equation with supercritical dis sipation and dispersive forcing in the whole space when the dispersive amplitude parameter is large enough we prove the global well posedness of strong solution to the equation with large initial data we also show the strong convergence result as the amplitude parameter goes to 1 both results rely on the strichartz type estimates for the corresponding linear equation

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

a priori estimates for the 3d quasi geostrophic system abstract the present article is devoted to the 3d dissipative quasi geostrophic system qg this system can be obtained as a limit model of the primitive equations in the asymptotics of strong rotation and stratification and involves a non radial non local homogeneous pseudo differential operator of order 2 denoted by and whose semigroup kernel reaches negative values after a refined study of the non local part of we prove a priori estimates in the general l p setting for the 3d qg model the main difficulty of this article is to study this highly non classical operator and the commutator of with a lagrangian change of variable an important application of these a priori estimates providing bound from below to the lifespan of the solutions of the primitive equations for ill prepared blowing up initial data can be found in a companion paper

global well posedness for a modified critical dissipative quasi geostrophic equation in this paper we consider the following modified quasi geostrophic equation t u d 0 u d 1r x r2 with 0 and 0 1 1 2 when 0 1 the equation was firstly introduced by constantin iyer and wu 2008 in 11 here by using the modulus of continuity method we prove the global well posedness of the system as a byproduct we also show that for every 0 2 the lipschitz norm of the solution has a uniform exponential upper bound

spatial sign cyclic feature detection this paper studies the impact of the spatial sign function on the cyclic features of communication signals and thereby on the performance of cyclic feature detectors it extends the existing results obtained for gaussian signals and provides new insights into the interactions between the cyclic features of a cyclostationary signal sent through a spatial sign filter which lead to the modified cyclic features at the output we apply these results to analytically derive the performance of the spatial sign cyclic correlation sscc detector which filters a received signal by the spatial sign function prior to its processing using a cyclic feature detector these results are extended to variations of the spatial sign function and we further motivate their application to general zero memory non linear zmnl functions

large time asymptotics of a fractional drift diffusion poisson system via the entropy method abstract the self similar asymptotics for solutions to the drift diffusion equation with fractional dissipation coupled to the poisson equation is analyzed in the whole space it is shown that in the subcritical and supercritical cases the solutions converge to the fractional heat kernel with algebraic rate the proof is based on the entropy method and leads to a decay rate in the l 1 r d norm the technique is applied to other semilinear equations with fractional dissipation

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

on hyperbolic dissipative systems of composite type abstract the shizuta kawashima condition plays the fundamental role in guaranteeing global stability for systems of hyperbolic parabolic hyperbolic with relaxation however there are many important physical systems not satisfying this coupling condition which are of composite type with regard to dissipation the compressible navier stokes equations with zero heat conductivity and euler equations of adiabatic flow through porous media are two typical examples in this paper we construct the global unique solution near constant equilibria to these two systems in three dimensions for the small h l l 3 initial data our proof is based on a reformation of the systems in terms of the pressure velocity and entropy a scaled energy estimates with minimal fractional derivative counts in conjunction with the linear l 2 l 2 decay estimates to extract a fast enough decay of velocity gradient which is used to close the energy estimates for the non dissipative entropy we also include an application to certain two phase models

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

a family of dissipative active scalar equations with singular velocity and measure initial data this paper considers a family of generalized active scalar equations with fractional dissipation whose velocity fields are more singular than riesz transform we prove global well posedness results for small initial data belonging to besov morrey spaces which contain strongly singular functions and measures concentrated at points diracs and on smooth curves self similar solutions are obtained for initial data and coupling velocity operator with correct homogeneities we also show an asymptotic behavior result and obtain a class of asymptotically self similar solutions

local and global strong solutions for sqg in bounded domains abstract we prove local well posedness for the inviscid surface quasigeostrophic sqg equation in bounded domains of r 2 when fractional dirichlet laplacian dissipation is added global existence of strong solutions is obtained for small data for critical and supercritical cases global existence of strong solutions with arbitrary data is obtained in the subcritical cases

stability of steady states of the compressible euler poisson system in r 3 abstract we investigate the global existence and asymptotic behavior of smooth solutions near a non flat steady state to the compressible euler poisson system in r 3 using some concise energy estimates and an interpolation trick we refine the global existence and show that the solution converges to the stationary solution exponentially fast we assume that the h 3 norms of the initial density and velocity are small but the higher derivatives can be arbitrarily large in this sense our method simplifies the proof of hsiao et al 2003 17 and improves the results of it

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

global existence and asymptotic stability of the fractional chemotaxis fluid system in r3 abstract in this paper we consider a model arising from biology consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in r 3 we establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument moreover the decay rates of the solutions and their higher order spatial derivatives are established towards the equilibrium by introducing the negative sobolev space h s 0 s 3 2

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

higher regularity of global attractors of a weakly dissipative fractional korteweg de vries equation in this paper we study the long time behavior of solutions to a weakly dissipative fractional korteweg de vries kdv equation on the real line r the main difficulty lies in that the dissipative term is a nonlocal operator we overcome this difficulty by the commutator estimates and product estimates associated with fractional laplacian the asymptotical compactness of solution semigroup is proved by the tail estimates finally we conclude the existence of h2 r h5 r global attractor of the weakly dissipative fractional kdv equation

large time behavior for the system of compressible adiabatic flow through porous media in r3 abstract the cauchy problem of the system of compressible adiabatic flow through porous media in r 3 is considered we obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in h 3 norm this is a continuing work of 31 where due to non dissipative property of the entropy s an additional assumption that the initial data is bounded in l 1 norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution in this paper to remove this assumption we first decompose the solution u into high frequency part u h and low frequency part u l then by constructing delicate energy functionals for u h and optimal decay rates for u l we can get the desired energy estimates to close the a priori assumption furthermore we get the optimal l 2 l 2 decay rate of the solution and show that the velocity decays faster than the density

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

finite time singularity for the modified sqg patch equation it is well known that the incompressible euler equations in two dimensions have globally regular solutions the inviscid surface quasi geostrophic sqg equation has a biot savart law which is one derivative less regular than in the euler case and the question of global regularity for its solutions is still open we study here the patch dynamics in the half plane for a family of active scalars which interpolates between these two equations via a parameter 0 12 appearing in the kernels of their biot savart laws the values 0 and 12 correspond to the 2d euler and sqg cases respectively we prove global in time regularity for the 2d euler patch model even if the patches initially touch the boundary of the half plane on the other hand for any sufficiently small 0 we exhibit initial data which lead to a singularity in finite time thus these results show a phase transition in the behavior of solutions to these equations and provide a rigorous foundation for classifying the 2d euler equations as critical department of mathematics rice university 6100 s main st houston tx 77005 1892 kiselev rice edu department of mathematics stanford university stanford ca 94305 ryzhik math stanford edu school of mathematics georgia institute of technology 686 cherry street atlanta ga 30332 0160 yaoyao math gatech edu department of mathematics university of wisconsin 480 lincoln dr madison wi 53706 1325 zlatos math wisc edu

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

asymptotic expansion of solutions to the nonlinear dissipative equation with the anomalous diffusion abstract the initial value problem for the dissipative equation with the extra force term is considered the dissipative effect on this equation is given by the fractional laplacian which describes the anomalous diffusion the main goal of this paper is to give an estimate on the difference between solutions and their asymptotic expansion as the space and the time variables tend to infinity furthermore as an application the large time behavior of solutions to some concrete problems are discussed

global existence and optimal convergence rates of solutions for 3d compressible magneto micropolar fluid equations abstract the cauchy problem for the three dimensional compressible magneto micropolar fluid equations is considered existence of global in time smooth solutions is established under the condition that the initial data are small perturbations of some given constant state moreover we obtain the time decay rates of the higher order spatial derivatives of the solution by combining the l p l q estimates for the linearized equations and the fourier splitting method if the initial perturbation is small in h 3 norm and bounded in l 1 norm

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 frequency localized maximum principle applied to the 2d quasi geostrophic equation in this paper we prove a maximum principle for a frequency localized transport diffusion equation as an application we prove the local well posedness of the supercritical quasi geostrophic equation in the critical besov spaces mathring b 1 alpha infty q and global well posedness of the critical quasi geostrophic equation in mathring b 0 infty q for all 1 q here mathring b s infty q is the closure of the schwartz functions in the norm of b s infty q

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

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

large time behavior for the system of a viscous liquid gas two phase flow model in r3 abstract the cauchy problem of a three dimensional compressible viscous liquid gas two phase flow model is considered in the present paper the global existence and uniqueness of solutions are established when the initial data is close to its equilibrium in the framework of sobolev space h 3 r 3 moreover the optimal l 2 l 2 convergence rates are also obtained for the solution

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

patterns of chaos synchronization small networks of chaotic units which are coupled by their time delayed variables are investigated in spite of the time delay the units can synchronize isochronally i e without time shift moreover networks cannot only synchronize completely but can also split into different synchronized sublattices these synchronization patterns are stable attractors of the network dynamics different networks with their associated behaviors and synchronization patterns are presented in particular we investigate sublattice synchronization symmetry breaking spreading chaotic motifs synchronization by restoring symmetry and cooperative pairwise synchronization of a bipartite tree

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

global well posedness for the critical dissipative quasi geostrophic equations in l abstract in this paper we study the critical dissipative quasi geostrophic equations in scaling invariant spaces we prove that there exists a global in time small solution for small initial data 0 l h 1 such that 0 l where is the riesz transform as a corollary we prove that if in addition 0 b q 0 1 q 2 is small enough then l t b q 0 l t 1 b q 1

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