Well-Posedness And Wave Breaking Analysis Of Solutions For Some Generalized Shallow Water Wave Type Equations

The motion of waves in water is one of the most remarkable phenomena in nature.Therefore,it is really essential to study the water movement,especially the water movement near the land surface.In this thesis,we mainly study the qualitative properties of solutions to the Cauchy problems for some generalized shallow water wave type equations related to tsunamis,tides and other phenomena,such as the well-posedness,wave breaking phenomena and blow-up rate.Firstly,we study the Cauchy problem of weakly dissipative generalized Novikov equation.First,by using Kato’s semi-group theory,the local well-posedness of solutions for the problem is obtained in Sobolev space H^s（R）,S>3/2.Second,by using the energy estimation,the wave breaking mechanism of solutions is established.In addition,it is proved that there are initial data which cause the wave breaking of corresponding solutions in finite time,and the blow-up rate of solutions is obtained when wave breaking occur.Finally,the global existence of solutions is obtained.Secondly,we study the Cauchy problem of weakly dissipative modified CamassaHolm equation in periodic case.First,by using Kato’s semi-group theory,the local well-posedness of solutions for weakly dissipative modified Camassa-Holm equation is obtained in Sobolev space H^t（S）,r>3/2.Second,by using the energy estimation,the wave breaking mechanism of solutions is established.And then it is proved that there is an initial datum which causes the wave breaking of the corresponding solution in finite time.In addition,the blow-up rate of solutions is obtained when wave breaking occur.Finally,the global existence of solutions is obtained.Then,we study the Cauchy problem of generalized Camassa-Holm equation.First,by using Kato’s semi-group theory,the local well-posedness of solutions for the problem is obtained in Sobolev space H^s（R）,s>3/2.Second,by using the energy estimation,the wave breaking mechanism of solutions is established.In addition,it is proved that there are initial data which cause the wave breaking of corresponding solutions in finite time,and the blow-up rate of solutions is obtained when wave breaking occur.Finally,the global existence of solutions is obtained.Finally,we study the global existence and uniqueness of solutions for the modified Camassa-Holm equation.First,by using the energy method,we establish the global existence and uniqueness of strong solutions for the problem in Sobolev space H^s（R）,s>3/2.Second,when initial datum u₀∈ H¹（R）and initial momentum density satisfies the sign condition,we use the compensated compactness method to mollify initial datum,and then use an approximation of global strong solutions with smooth initial data to obtain the existence of the global weak solution.Finally,by using the regularization technique,the uniqueness of global weak solutions is proved.