摘要
The local well-posedness for the b-family equation in the critical space is studied.Applying the Littlewood-Paley decomposition method in the critical Besov spaces Bs2r with the index s=3/2 which is the generalization space of the Sobolev spaces Hs it is established that there is a maximal time T=Tu0＆gt;0 such that for the b-family equation there exists a unique solution utx∈C0TB3/221∩C10TB121 when the initial variable u0x ∈B3/221 is the critical regularity. Moreover the solution uxt depends continuously on the initial data u0x .Furthermore using the abstract Cauchy-Kowalevski theorem to prove the analytic regularity of the solutions for the b-family equation in a suitable scale of Banach spaces E it is shown that the solutions of the b-family are analytic in both variables globally in space and locally in time when the initial data is analytic.
首先研究了b-family方程在临界空间中的局部适定性.在参数为s=3/2的临界Besov空间Bs2,r(该空间是Sobolev空间Hs的一种推广形式)中,采用Littlewood-Paley分解方法,得到当初值u0(x)∈B3/22,1为临界正则时,存在最长时间T=T(u0)>0,使得b-family方程有唯一解u(t,x)∈C[0,T];B3/22,()1∩C1([0,T];B12,1),且解u(x,t)是连续依赖于初值u0(x).进一步,在合适的Besov尺度空间E中,运用抽象的CauchyKow alevski定理研究b-family方程解的解析性,证得:当初值是解析的,则该方程解在全空间和局部时间内也是解析的.
基金
The Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No.13KJB110012)
