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

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

global regularity for a class of 2d generalized tropical climate models abstract this paper establishes the global existence and regularity of solutions to a two dimensional 2d tropical climate model tcm with fractional dissipation the inviscid counterpart of this model was derived by frierson majda and pauluis 8 as a model for tropical geophysical flows this model reflects the interaction and coupling among the barotropic mode u the first baroclinic mode v of the velocity and the temperature the systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization the aim here is the global regularity with the least dissipation we prove two main results first the global regularity of the system with v and for 1 and 3 2 and second the global regularity of the system with v for 3 2 the proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal the key tools employed here include the maximal regularity for general fractional heat operators the littlewood paley decomposition and besov space techniques lower bounds involving fractional laplacian and simultaneous estimates of several coupled quantities

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

existence of solutions to the convection problem in a pseudomeasure type space we study the well posedness of a convection problem in a pseudomeasure type space pma without assuming that the gravitational field is bounded considering the pma space with right homogeneity th

the well posedness of the surface quasi geostrophic equations in the besov morrey spaces abstract in this paper we develop the boundedness of riesz transform in morrey spaces which generalize the corresponding result in the usual l p spaces as an application of this boundedness we establish the well posedness and blow up criterion for the surface quasi geostrophic equations in the framework of besov morrey spaces

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

on the quasi geostrophic equations on compact closed surfaces in r3 abstract the quasi geostrophic equations on compact surfaces in r 3 without boundary are studied by means of the theory of quasi linear parabolic evolution equations based on maximal l p regularity it is shown that the problem is globally strongly well posed in l q the solutions regularize instantly and form a global semiflow in the proper state manifold and as time goes to infinity a solution converges exponentially to a constant in the topology of the state space

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

weighted lp estimates and fujita exponent for a nonlocal equation abstract we investigate a nonlocal equation t u r n j x y u y t d y j l 1 u x t a x t u p in r n where a is unbounded and j belongs to a weighted space crucial weighted l p and interpolation estimates for the green operator are established by a new method based on the sharp young s inequality the asymptotic behavior of a regular varying coefficients exponential series and the properties of auxiliary functions 1 x 2 b 2 that j and b 2 x b b 2 blow up behaviors are investigated by employing test functions r r instead of principal eigenfunctions global well posedness in weighted l p spaces for the cauchy problem is proved when a x the fujita exponent is shown to be 1 2 n our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations

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

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

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

pointwise estimates for solutions of fractal burgers equation abstract in this paper we provide two sided estimates and uniform asymptotics for the solution of d dimensional critical fractal burgers equation u t 2 u b u u q 0 1 2 b r d for q 1 d and u 0 l 1 r d we consider also q 1 d under additional condition u 0 l r d in both cases we assume u 0 0 which implies that the solution is non negative the estimates are given in the terms of the function p t u 0 where p t denotes the semigroup for the operator t 2

global well posedness for a modified dissipative surface quasi geostrophic equation in the critical sobolev space h1 abstract in this paper we consider the following modified quasi geostrophic equations mqg t u 0 u 1 r where 0 1 is a fixed parameter this equation was recently introduced by p constantin g iyer and j wu 2001 in 4 as a modification of the classical quasi geostrophic equation in this paper we prove that for any initial data in the sobolev space h 1 r 2 eq mqg has a global and smooth solution in c r h 1 r 2

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 leitmann schmitendorf advertising differential game the paper revisits the advertising differential game suggested by leitmann and schmitendorf 1 we put the model into perspective and discuss the related issues of i state separable or linear state games and ii open loop equilibria that are feedback equilibria which are constant with respect to state

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 decomposition method for large scale milps with performance guarantees and a power system application lagrangian duality in mixed integer optimization is a useful framework for problem decomposition and for producing tight lower bounds to the optimal objective however in contrast to the convex case it is generally unable to produce optimal solutions directly in fact solutions recovered from the dual may not only be suboptimal but even infeasible in this paper we concentrate on large scale mixed integer programs with a specific structure that appears in a variety of application domains such as power systems and supply chain management we propose a solution method for these structures in which the primal problem is modified in a certain way guaranteeing that the solutions produced by the corresponding dual are feasible for the original unmodified primal problem the modification is simple to implement and the method is amenable to distributed computation we also demonstrate that the quality of the solutions recovered using our procedure improves as the problem size increases making it particularly useful for large scale problem instances for which commercial solvers are inadequate we illustrate the efficacy of our method with extensive experimentations on a problem stemming from power systems

on the global well posedness of 2 d boussinesq system with variable viscosity abstract in this paper we investigate the global well posedness of 2 d boussinesq system which has variable kinematic viscosity and with thermal conductivity of d with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in l norm

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

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

regularity criterion for a critical fractional diffusion model of two dimensional micropolar flows abstract this paper is devoted to the beale kato majda bkm regularity criterion of the 2d micropolar equations in the critical fractional diffusion situation 1 for 0 and 0 with the use of the littlewood paley theory the global existence of the unique solution u w to the equations is found to be completely determined by the velocity field u the dynamical behaviour of the scalar angular velocity w is controlled by the velocity u more precisely it is shown that the strong solution u w exists globally if 0 t u l 2 d for any t

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

heuristic concentration two stage solution construction abstract by utilizing information from multiple runs of an interchange heuristic we construct a new solution that is generally better than the best local optimum previously found this new two stage approach to combinatorial optimization is demonstrated in the context of the p median problem two layers of optimization are superimposed the first layer is a conventional heuristic the second is a heuristic or exact procedure which draws on the concentrated solution set generated by the initial heuristic the intention is to provide an alternative heuristic procedure which when dealing with large problems has a higher probability of producing optimal solutions than existing methods the procedure is fairly general and appears to be applicable to combinatorial problems in a number of contexts

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

on the two dimensional muskat problem with monotone large initial data we consider the evolution of two incompressible immiscible fluids with different densities in porous media known as the muskat problem 21 which in two dimensions is analogous to the hele shaw cell 24 we establish for a class of large and monotone initial data the global existence of weak solutions the proof is based on a local well posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function 2016 wiley periodicals inc

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

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

on a 1d nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion abstract we study a 1d transport equation with nonlocal velocity with subcritical or supercritical dissipation for all data in the weighted sobolev space h k w l where k max 0 3 2 and w is a given family of muckenhoupt weights we prove a global existence result in the subcritical case 1 2 we also prove a local existence theorem for large data in h 2 w l in the supercritical case 0 1 the proofs are based on the use of the weighted littlewood paley theory interpolation along with some new commutator estimates

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

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

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

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

on the global evolution of vortex filaments blobs and small loops in 3d ideal flows we consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions the proof exploits the existence of suitable hamiltonian functions the approximate models we analyze essentially discrete and continuous vortex filaments and vortex loops are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high reynolds number finally we apply our strategy to discrete models for filaments used in numerical methods

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

effective heuristics for multiproduct partial shipment models motivated by real applications we consider the problem of shipping products to multiple customers from limited inventory after formulating the optimization problems under different restrictions on partial shipments we find that commercially available packages applied directly are unsatisfactory as are simple greedy approaches we develop a scheme of heuristics that enables the user to select a good balance between computation time and effectiveness a detailed computational study of one and two period industrial sized problems indicates that these heuristics are computationally practical and generate solutions that are on average within 34 of the optimum

age structured linear state differential games abstract in this paper we search for conditions on age structured differential games to make their analysis more tractable we focus on a class of age structured differential games which show the features of ordinary linear state differential games and we prove that their open loop nash equilibria are sub game perfect by means of a simple age structured advertising problem we provide an application of the theoretical results presented in the paper and we show how to determine an open loop nash equilibrium

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

existence and decay of solutions to the dissipative quasi geostrophic equation with delays abstract we consider the dissipative quasi geostrophic equation with forcing and advective terms containing delays we prove the existence of weak solutions and algebraic decay of the energy provided the initial data 0 is in l p l 2 1 p 2 and the forcing term satisfies some mild conditions on its long time behavior

exact and heuristic solutions for a shipment problem with given frequencies we consider the problem of shipping several products from an origin to a destination when a discrete set of shipping frequencies is available in such a way that the sum of the transportation and inventory costs is minimized this problem which is known to be np hard has applications in transportation planning and in location analysis in this paper we derive dominance rules for the problem solutions that allow a tightening of the bounds on the problem variables and improve the efficiency of a known branch and bound algorithm moreover we present some heuristics and compare them with two different modifications of an eoq type algorithm for the solution of the problem with continuous frequencies

on the 2d critical and supercritical dissipative quasi geostrophic equation in besov spaces abstract we prove the local smoothing effect of the 2d critical and supercritical dissipative quasi geostrophic equations in critical besov spaces as an application a global well posedness result is established by adapting a method in kiselev nazarov and volberg 2007 16 and an idea in dong and du 2008 15 with suitable modifications moreover we show that the unique solution obtained in chen miao and zhang 2007 11 is a classical solution these generalize some previous results in dong 2010 13 dong and du 2008 15 the main ingredients of the proofs are two commutator estimates and the preservation of suitable modulus of continuity of the solutions

the nash solution of an advertising differential game generalization of a model by leitmann and schmitendorf in this paper the differential game between two profit maximizing firms considered in 15 is extended to the case of general concave effectiveness functions of advertising for this model a phase portrait analysis of nash equilibrium solutions is carried out providing qualitative insights into the structure of optimal advertising rates applying this analysis to the leitmann schmitendorf advertising model we obtain more detailed information on the structure of the optimal controls moreover nash optimal solutions are derived for infinite duration of the game

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

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

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

solutions of the 2d quasi geostrophic equation in holder spaces abstract the 2d quasi geostrophic equation t u 0 u r is a two dimensional model of the 3d hydrodynamics equations when 1 2 the issue of existence and uniqueness concerning this equation becomes difficult it is shown here that this equation with either 0 or 0 and 0 1 2 has a unique local in time solution corresponding to any initial datum in the space c r l q for r 1 and q 1