Low Regularity And Related Harmonic Analysis Of Some Kdv Type Equations

This paper mainly studies the related results of the rotation-modified KadomtsevPetviashvili(RMKP)type equation and the free Ostrovsky equation of the two types of dispersion equations.The first chapter mainly studies the Cauchy problem for the rotation-modified KadomtsevPetviashvili(RMKP)type equation ut-?(?)x3u+(?)x(u2)+?'(?)x-1(?)y2u-?(?)x-1=0(0-2)in anisotropic Sobolev space Hs1,s2(R2).In this chapter,the main difficulty is how to deal with the rotation term y(?)x-1u in establishing Strichartz estimates and bilinear estimates.Theorem 2.1.1.(Bilinear estimate)Let s1>-1/2,s2?0 and b=1/2,b'=-1/2+?,?=1/2+?,Then we have We prove that the Cauchy problem for the above equation is locally well-posed in the anisotropic Sobolev spaces Hs1,s2(R2)with s1>-1/2and s2?0,?=?=1=-?'.This theorem can be stated as follows:Theorem 2.1.2.(Local well-posedness)Let|?|-1Fxyu0(?,?)?S'(R2),s1>-1/2,s2?0.For R>0,there exists T=T(R)>0 and a Banach space(?)(?)Hsl,s2(R2))such that for every ?BR:=?u0?Hs1,s2(R2)|?u0?Hs1,s2(R2)0 for which the map(?)t?[0,T]is C3 at zero from Hs1,0(R2)to Hs1,0(R2).In the second chapter,we mainly prove that a certain initial value f of pointwise convergence of linear Ostrovsky equation in Hs(R)(s?1/4)and the problem of stochastic continuity of random initial value fw in L2(R).We obtain the following theorems.Theorem 3.1.1.(Pointwise convergence)Let f?Hs(R)(s?1/4).Then we have almost everywhere with respect to x.At the same time,counterexample is constructed to show that the maximal function estimate related to the free Ostrovsky equation can fail if s<1/4.We get the following theorem results.Theorem 3.1.2.For(?)we have The inequality?U(t)fk?Lx4Lt???fk?Hs(R)does not hold.Finally,we show the stochastic continuity at t=0 of free Ostrovsky equation with random data in L2(R).We get the following theorem.Theorem 3.1.3.(Stochastic continuity)Letf?L2(R)and fwbe a randomization of f.Then(?)we have More precisely,(?)such that 2Cee(ln 3C1/?)1/2?})??.Here C,C1 are contants that do not independent x,t,?.