| 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-β（?）x3u+（?）x（u2）+β’（?）x-1（?）y2u-γ（?）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+∈,σ=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,β=γ=1=-β’.This theorem can be stated as follows:Theorem 2.1.2.（Local well-posedness）Let|ξ|-1Fxyu0（ξ,μ）∈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<α and when |t|<∈2,we have P（{w∈Ω:|U（t）fw-fw|>α}）≤∈.Here C,C1 are contants that do not independent x,t,∈. |
