Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces

We study the Cauchy problem of a two-species chemotactic model. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish a unique local solution and blow-up criterion of the solution, when the initial data(u0, v0, w0) belongs to homogeneous Besov spaces˙B~(-2+3/p)_(p,1)(R~3) ×˙B~(-2+3/r)_(r,1)(R~3) ×˙B~(3/q)_(q,1)(R~3) for p, q and r satisfying some technical assumptions. Furthermore, we prove that if the initial data is sufficiently small, then the solution is global. Meanwhile, based on the so-called Gevrey estimates, we particularly prove that the solution is analytic in the spatial variable. In addition, we analyze the long time behavior of the solution and obtain some decay estimates for higher derivatives in Besov and Lebesgue spaces.     

Wellposedness of some quasi-linear Schrdinger equations

This article is devoted to the study of a quasilinear Schrdinger equation coupled with an elliptic equation on the metric g. We first prove that, in this context, the propagation of regularity holds which ensures local wellposedness for initial data small enough in˙H1/2 and belonging to the Besov space˙B3/22,1. In a second step, we establish Strichartz estimates for time dependent rough metrics to obtain a lower bound of the time existence which only involves the˙B1+ε2,∞norm on the initial data.     

本刊英文版2015年58卷第5期摘要（英文）

<正>Wellposedness of some quasi-linear Schrodinger equations CHEMIN Jean-YvesSALORT Delphine Abstract This article is devoted to the study of a quasilinear Schrodinger equation coupled with an elliptic equation on the metric g.We first prove that,in this context,the propagation of regularity holds which ensures local wellposedness for initial data small enough in H~(1/2)and belonging to the Besov space B_(2,1)~(3/2).In a second step,we establish Strichartz estimates for time dependent rough metrics to obtain a lower bound of the time existence     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.     

GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON R3×T WITH CUBIC NONLINEARITIES

In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R³× T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L²-L~∞ decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with...     

Well-posedness for the 2D dissipative quasigeostrophic equations in the Besov space

In this paper, we consider the initial value problem of the 2D dissipative quasi-geostrophic equations. Existence and uniqueness of the solution global in time are proved in the homogenous Besov space Bp,∞ s p with small data when 1 /2<α≤1,2/2α-1< p<∞,sp=2/p-(2α-1). Our proof is based on a new characterization of the homogenous Besov space and Kato's method.     

GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA

We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG) in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP) at the asymptotic rate of 1/c² as the speed of light c tends to infinity for al...     

ASYMPTOTIC BEHAVIOR OF THE NONLINEAR PARABOLIC EQUATIONS

This paper is concerned with the large time behavior for solutions of the nonlinear parabolic equations in whole spaces R^n. The spectral decomposition methods of Laplace operator are applied and it is proved that if the initial data u0∈ L^2 ∩ L^r for 1 ≤ r ≤ 2, then the solutions decay in L^2 norm at t^-n/2(1/r-1/2). The decay rates are optimal in the sense that they coincide with the decay rates of the solutions to the heat equations with the same initial data.     

The Cauchy Problem of the Nonlinear Schr■dinger Equations in R~(1 1)

In this paper,we consider the local and global solution for the nonlinear Schrdinger equationwith data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data.Thetechniques to be used are adapted from the Strichartz type estimate,Kato's smoothing effect and the maximalfunction(in time)estimate for the free Schrdinger operator.     

Time Decay Rates of the isotropic Non-Newtonian Flows in R^n

This paper is concerned with time decay rates for weak solutions to a class system of isotropicincompressible non-Newtonian fluid motion in R~n.With the use of the spectral decomposition methods ofStokes operator,the optimal decay estimates of weak solutions in L~2 norm are derived under the differentconditions on the initial velocity.Moreover,the error estimates of the difference between non-Newtonian flowand Navier-Stokes flow are also investigated.     

The Cauchy Problem of the Nonlinear Schrodinger Equations in R^1＋1

In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.     

Global well-posedness for the 2D critical dissipative quasi-geostrophic equation

We study the initial value problem for the 2D critical dissipative quasi-geostrophic equation. We prove the global existence for small data in the scale invariant Besov spaces Bp,12/p,1≤p≤∞. In particular, for p=∞, our result does not impose any regularity on the initial data. Our proofs are based on an exponential decay estimate of the semigroup e-tk(-Δ)αand the use of space-time Besov spaces.     

大初值的抛物型守恒律方程解的大时间性态(英文)

In this paper,we study the large-time behavior of periodic solutions for parabolic conservation laws.There is no smallness assumption on the initial data.We firstly get the local existence of the solution by the iterative scheme,then we get the exponential decay estimates for the solution by energy method and maximum principle,and obtain the global solution in the same time.     

Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions

This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t→-∞in the energy norm, and to show it has a free profile as t→ ∞. Our approach is based on the work of [11]. Namely we use a weighted L∞norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.     

Large time behaviors of the isentropic bipolar compressible Navier-Stokes-Poisson system

The isentropic bipolar compressible Navier-Stokes-Poisson (BNSP) system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data belong to the Sobolev space H l(R3) ∩ B˙ s 1,1 (R3) with l ≥ 4 and s ∈ (0, 1], it is shown that the momenta of the charged particles decay at the optimal rate (1+t) 1 4 s 2 in L2 -norm, which is slower than the rate (1+t) 3 4 s 2 for the compressible Navier-Stokes (NS) equations [14]. In particular, a new phenomenon on the charge transport is observed. The time decay rate of total density and momentum was both (1 + t) 3 4 due to the cancellation effect from the interplay interaction of the charged particles.     

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

ASYMPTOTIC BEHAVIOR OF THE STOKES APPROXIMATION EQUATIONS FOR COMPRESSIBLE FLOWS IN R~3

We consider the Stokes approximation equations for compressible flows in R3.The global unique solution and optimal convergence rates are obtained by pure energy method provided the initial perturbation around a constant state is small.In particular,the optimal decay rates of the higher-order spatial derivatives of the solution are obtained.As an immediate byproduct,the usual L~p-L~2(1≤p≤2) type of the optimal decay rate follow without requiring that the L~p norm of initial data is small.