Well-posedness Of A Kind Of The Free Surface Equation Of Shallow Water Wave

In this paper,we consider the well-posedness of the free surface equation,which approximates solutions consistent with the GN equation(see[1]).The family of equations?t+?x+3/2???x-3/8?2?2?x+3/16?3?3?x+?(??xxx + ??xxt)= ??(???xxx???x?xx),(0.4)can be used to construct an approximate solution consistent with the GN equa-tions,where x ???(t,x)parameterizes the elevation of the free surface at time t,?,?,?,?,? and ? are constants.Let q ? R and assume that?=q,?=q-1/6,?=-3/2q-1/6,?=-9/2q-5/24especially,choosing q = 1/12,??12,?=1 the equation(0.4)reads?t+?x+3/2??x-3/8?2?x+3/16?3?x+?xxx-?xxt=-7/2??xxx-7?x?xx.(0.5)We get the equivalent form of the free surface equation(0.5)and initial condition:{(?)?-(?)x?-7/2?(?)x?=(1-(?)2x)-1(?)x(-2?-5/2?2+7/4((?)x?)2+1/8?3-3/64?4),(?)t>0,x ? R,?|t=0=?0,(?)x ? R.(0.6)This paper is concerned with the Cauchy problem of the free surface equation,that is the model(0.6),applying some basic inequalities,commutator estimation and transport theorem in Sobolev space to prove the local well-posedness of this equation.On this basis,we obtain the wave-breaking conditions of this system.The thesis has four chapters,which are arranged as follows:At first,we briefly describe the background of the Shallow water equation and the GN equation,and also the progress of the system mentioned both at home and abroad.Secondly,we illustrate related notations and definitions.Next we list the lemmas and corollaries during the course of the proof.Thirdly,we establish the local well-posedness of the Cauchy problem of the free surface system(0.6).At last,we give the wave-breaking conditions of(0.6).