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 . We prove local existence of a unique regular solution for arbitrary initial data in H ^2-2α 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.     

Nonlinear Instability for the Critically Dissipative Quasi-Geostrophic Equation

We prove that linear instability implies non-linear instability in the energy norm for the critically dissipative quasi-geostrophic equation.     

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

 We consider the quasi-geostrophic equation with the dissipation term, κ (-Δ)^α θ, 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 ^2−2α _2,1 , for initial data small in the B ^2−2α _2,1 norm. We also prove a global stability result in B ¹ _2,1 . Received: 24 April 2002 / Accepted: 29 July 2002 Published online: 10 December 2002 Communicated by P. Constantin     

Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation

We study the evolution of sharp fronts for the Surface Quasi-Geostrophic equation in the context of analytic functions. We showed that, even though the equation contains operators of order higher than 1, by carefully studying the evolution of the second derivatives it can be adapted to fit an abstract version of the Cauchy-Kowaleski Theorem.     

Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ₀ is in L ² only, we prove that the L ² norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ₀ in L ^p ∩ L ², with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L ². We also prove that when the norm of θ₀ is small enough, the L ^q norms, for , have uniform decay rates. This result allows us to prove decay for the L ^q norms, for , when θ₀ is in . The second author was partially supported by NSF grant DMS-0600692.     

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 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 s0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.Acknowledgement The author thanks Prof. P. Constantin for encouragement and kind help for his research on the subject of 2D QG equations, Prof. J. Wu for useful conversation and Prof. A. Cordorba for providing preprints. This work was started while the author visited IPAM at UCLA with an IPAM fellowship. The hospitality and support of IPAM is gratefully acknowledged. This work is partially supported by the Oklahoma State University new faculty start-up fund and the Deans Incentive Grant.     

Blow Up for the Generalized Surface Quasi-Geostrophic Equation with Supercritical Dissipation

We prove the existence of singularities for the generalized surface quasi-geostrophic (GSQG) equation with supercritical dissipation. Analogous results are obtained for the family of equations interpolating between GSQG and 2D Euler.     

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 Lemarié-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.     

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 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 s2(1–) and the initial data is small. For the case, s>2(1–), we also obtain the unique large global solution in H^s provided that is small enough.Acknowledgement The author thanks Professor Jiahong Wu for useful conversations, Professor Antonio Cordoba for kindly providing their preprints and Professor Peter Constantin for kind suggestions and encouragement. This work is partially supported by the Oklahoma State University, School of Art and Science new faculty start-up fund and by the Deans Incentive Grant.     

Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation

This paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the Degasperis-Procesi equation on the line. Global strong solutions to the equation are determined for a class of initial profiles. On the other hand, it is shown that the first blow-up can occur only in the form of wave-breaking. A new wave-breaking mechanism for solutions is described in detail and two results of blow-up solutions with certain initial profiles are established.     

Existence of Global Weak Solutions for a 2D Viscous Shallow Water Equations and Convergence to the Quasi-Geostrophic Model

 We consider a two dimensional viscous shallow water model with friction term. Existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi-geostrophic equation with free surface term is proven in the well prepared case. The ill prepared data case is also discussed. Received: 4 October 2002 / Accepted: 22 January 2003 Published online: 28 May 2003 Communicated by P. Constantin     

Global Existence Proof for Relativistic Boltzmann Equation with Hard Interactions

By combining the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation and the Dudyński and Ekiel-Jeżewska device of the causality of the relativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic hard interactions and the initial data satisfying finite mass, energy and entropy. This is in fact an extension of the result of Dudyński and Ekiel-Jeżewska to the case of the relativistic Boltzmann equation with hard interactions. This work was supported by NSFC 10271121 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China, and sponsored by joint grants of NSFC 10511120278/10611120371 and RFBR 04-02-39026.     

Dissipative Linear Boltzmann Equation for Hard Spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard-spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non-zero temperature lower than the background one. Moreover thanks to the H-Theorem we prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

Global Regularity for the Maxwell-Klein-Gordon Equation with Small Critical Sobolev Norm in High Dimensions

We show that in dimensions n 6 one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [17, 7, 14, 12] but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.Acknowledgements The authors are deeply indebted to Sergiu Klainerman, without whose encouragement and insight this project would not even have been initiated. We have tremendously benefited from numerous discussions with him, in particular on the issue of the Cronstrom gauge. We would also like to thank Joachim Krieger and the anonymous referee for their helpful comments and suggestions.I.R. is a Clay Prize Fellow and supported in part by the NSF grant DMS-01007791T.T. is a Clay Prize Fellow and supported in part by a grant from the Packard Foundation     

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 \mathringB^1-a_￥,q{\mathring{B}^{1-\alpha}_{\infty,q}}, and global well-posedness of the critical quasi-geostrophic equation in \mathringB⁰_￥,q{\mathring{B}^{0}_{\infty,q}} for all 1 ≤ q < ∞. Here \mathringB^s_￥,q {\mathring{B}^{s}_{\infty,q} } is the closure of the Schwartz functions in the norm of B^s_￥,q{B^{s}_{\infty,q}}.     

Dissipative Solitons in the Complex Ginzburg-Landau Equation for Femtosecond Lasers

We consider the models of generating femtosecond radiation in solid-state lasers with passive mode locking. An approach in which ultrashort laser pulses (USPs) are associated with the complex Ginzburg-Landau equation (GLE) is developed. The structure of solitons and the range of parameters of their existence are analytically determined and the stability criteria of those solitons are formulated for a number of limiting cases. The scenarios of the cyclic spectral and temporal dynamics of dissipative solitons due to the instability of the background field are studied.