三維可壓縮Euler方程組經典解的爆破

朱旭生,李芳娥,俞銀晶

（華東交通大學基礎科學學院,江西南昌 330013）

關于三維可壓縮Euler方程組[1-2]研究成果已有很多,主要集中在各種形式的弱解以及經典解的爆破[3-8],其結論是在某些指標數據較大時經典解必定在有限時間內爆破;在有阻尼的情形下獲得小初值時的經典解的整體存在性,例如Sideris TC研究了三維可壓縮歐拉方程組解的奇異性的形成,即在某些初始數據較大的情形下經典解的爆破,以及王維克等在初始數據較小時得到了帶阻尼項的多維可壓縮歐拉方程組經典解的整體存在性,并研究了解的點態估計等。本文繼續研究三維空間中等熵Euler方程組的初值問題,在Sideris TC研究了三維空間中可壓縮歐拉方程組經典解爆破的基礎上,對條件進行適當的調整,結合文獻[6-7],通過構造泛函,證明其經典解在有限時間內必定爆破的結論。

結論及證明

考慮三維等熵可壓縮Euler方程組:

[1]柯朗,弗里德里克斯.超聲速流與沖擊波[M].李維新,等,譯.北京:科學出版社,1986.

[2]MAJDA A.Compressible fluid flow and systems of conservation laws in several space variables[M].New York:Springer-Verlag,1984.

[3]SIDERIS T C.Formation of singularities in three dimensional compressible fluids[J].Comm.Math.Phys.,1985,101（4）:475-485.

[4]LIU T P.The development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations[J].J Differential Equations,1979,33（2）:92-111.

[5]SIDERIS T C.The life span of smooth solutions to the three dimensional compressible Euler equations and the incompressible limit[J].Indiana Univ Math,1991,40（2）:535-550.

[6]ALINHAC S.Blowup for nonlinear hyperbolic equations[M].Bostn,Birkhauser:1995:33-36.

[7]XIONGFENG YANG,WEIKE WANG.The suppressible property of the solution for three-dimensional Euler equations with damping[M].Nonlinear Analysis,2007,8（1）:53-61.

[8]ZHU XUSHENG,WANG WEIKE.The regular solutions of the isentropic Euler equations with degenerate linear damping[J].Chin Ann Math,2005,26B（4）:583-598.

[9]WANG WEIKE,YANG TONG.The pointwise estimates of solutionsfor Euler equationswith damping inMulti-Dimensions[J].Journal of Differential Equations,2001,173（2）:410-450.

[10]SIDERIS T C,THOMASES B,WANG DEHUA.Long time behavior of solutions to the 3D compressible Euler equations with damping[J].Comm.Partial Diff.Eqns.,2003,28（3&4）:795-816.