三維可壓縮Euler方程組經(jīng)典解的爆破

朱旭生,李芳娥,俞銀晶

（華東交通大學(xué)基礎(chǔ)科學(xué)學(xué)院,江西南昌 330013）

關(guān)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(jīng)典解的爆破[3-8],其結(jié)論是在某些指標(biāo)數(shù)據(jù)較大時(shí)經(jīng)典解必定在有限時(shí)間內(nèi)爆破;在有阻尼的情形下獲得小初值時(shí)的經(jīng)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(shù)據(jù)較大的情形下經(jīng)典解的爆破,以及王維克等在初始數(shù)據(jù)較小時(shí)得到了帶阻尼項(xiàng)的多維可壓縮歐拉方程組經(jīng)典解的整體存在性,并研究了解的點(diǎn)態(tài)估計(jì)等。本文繼續(xù)研究三維空間中等熵Euler方程組的初值問(wèn)題,在Sideris TC研究了三維空間中可壓縮歐拉方程組經(jīng)典解爆破的基礎(chǔ)上,對(duì)條件進(jìn)行適當(dāng)?shù)恼{(diào)整,結(jié)合文獻(xiàn)[6-7],通過(guò)構(gòu)造泛函,證明其經(jīng)典解在有限時(shí)間內(nèi)必定爆破的結(jié)論。

結(jié)論及證明

考慮三維等熵可壓縮Euler方程組:

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