二维等熵可压欧拉方程古典解的存在性(英文).doc

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1、1二维等熵可压欧拉方程古典解的存在性(英文)AbstractIn this paper, the authors study the local existence of classical solution of the 2D isentropic compressible Euler equation, by using the iterative approach, the local existence and uniqueness is obtained, and also proved that the solution blow up infinite time, that is

2、, there is no global classical solution for compressible Euler equation. Key wordsIsentropic compressible Euler equations; Local existence; Blow-up criterion CLC numberO 175Document codeA 1Introduction In this paper, we consider the 2D isentropic compressible Euler equations as follow: The Euler equ

3、ations is used to describe the perfectfluids which corresponds to the particular case of Navier-Stokes equations. The Navier-Stokes equations for isentropic compressibleflow in two dimension can be express in the form rnal force, the viscosity coefficientsandsatisfy 0, 2Many results concerning the l

4、ocal existence of equations (3) can be found in 1-4 when0 0. There are also some local existence results in 5-7 when the initial density is nonnegative. For the blow-up criterion problem, refer for instance to 8-11 and references therein. For incompressible case, Schaeffer12and McGrath13researched t

5、he Euler equations in R2. In 14, Temam obtained the local existence of classical solution of Euler equations. Motivated by 12,14, we consider the local existence of classical solution of the 2D isentropic compressible Euler equation. Our main results are formulated as following theorems: Theorem 1As

6、sume that (0,u0)Hs(R2)Hs(R2) for some s 2. Then there exists a unique local classical solution (,u)C(0,T) ;Hs(R2)Hs(R2) ) to the Cauchy problem (1)-(2) , for some T = T(0Hs(R2) ,u0Hs(R2) ). The rest of the paper is organized as follows: In Section 2, we state some elementary facts and inequalities w

7、hich will be needed in later analysis. Section 3 gives out the proof of Theorem 1. Section 4 gives out the proof of Theorem 2. Step 3Continuity and uniqueness of solutions. By view of (31) , one can deduce thatk, ukconverge 3to, u in C(0,T?;Hs?1) , C(0,T?;Hs?1) as k+, respectively. From the solution

8、 of (8) (9) , it is easy to know that (,u) is a solution of (1) (2) , and belongs to C(0,T?;Hs) , C(0,T?;Hs) , respectively. Therefore, the proof of existence is completed. Finally, we prove the uniqueness of local solution. Assume that (1,u1) and(2,u2) are both two solutions of the problem (1) (2)

9、, then we can use the same method as Step 2, and also obtain similar estimate (31) , so we obtain the proof of uniqueness. Then we obtain the proof of Theorem 1. 1 N Itaya. On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscousfluids. Kodai Math

10、 Sem Rep, 1971, 23: 60-120. 2 N Itaya. On the initial value problem of the motion of compressible viscousfluid, especially on the problem of uniqueness. J Math Kyoto Univ, 1976, 16: 413-427. 3 J Nash. Le problme de Cauchy pour lesquations differentielles dunfluid gnral. Bull Soc Math France, 1962, 9

11、0: 487-497. 4 A Tani. On thefirst initial-boundary value problem of compressible viscousfluid motion. Publ Res Inst Math Sci Kyoto Univ, 1971, 13: 193-253. 45 Y Cho, H J Choe, H Kim. Unique solvability of the initial boundary value problems for compressible viscousfluid. J Math Pures Appl, 2004, 83:

12、 243-275. 6 Y Cho, H Kim. Strong solutions of the Navier-Stokes equations for isentropic compressiblefluids. J DiffE, 2003, 190: 504-523. 7 Y Cho, H Kim. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscripta Math, 2006, 120: 91-129. 8 D Y

13、Fang, R Z Zi, T Zhang. A blow-up criterion for two dimensional compressible viscous heat-conductiveflows. Non Anal, 2012, 75(6): 3130-3141. 9 X D Huang, Z P Xin. A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. Sci C Math, 2010, 53(3): 671-686. 10 X D Huang, J

14、 Li, Z P Xin. Blowup criterion for viscous baratropicflows with vacuum states. Comm Math Phy, 2011, 301(1): 23-35. 11 Y Z Sun, C Wang, Z F Zhang. A Beale-Kato-Majda blow-up criterion for 3-D compressible Navier- Stokes equations. J Math Pures A, 2011, 95(1): 36-47. 12 A C Schaeffer. Existence theore

15、m for theflow of an 5incompressiblefluid in two dimensions. Trans Amer Math Soc, 1937, 42:, 497-513. 13 F J McGrath. Nonstationary planeflow of viscous and idealfluids. Arch Rational Mech Anal, 1968, 27: 329-348. 14 R Temam. Local existence of Csolutions of the Euler equations of incompressible perf

16、ectfluids. In: Lecture Notes in Mathematics, Vol 565, Berlin, Heidelberg, New York: Springer, 1976, 184-194. 15 M Taylor. Pseudodifferential operators and nonlinear PDE. Birkhauser, Boston, 1991. 16 X J Xu. Local existence and blow-up criterion of the 2-d compressible Boussinesq equations without di

17、ssipation terms. Dis Cont Dynam Sys, 2009, 25(4): 1333-1347. 17 T Tao. Multilinear weighted convolution of L2-functions, and application to nonlinear dispersive equations. Amer J Math, 2001, 123: 839-908. 18 T Kato. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch Ration Mech Anal, 1975, 58: 181-205.

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