简谈英文二维等熵可压欧拉方程古典解有着性(英文)
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论文导读：quation.Ourmainresultsareformulatedaollowingtheorems：Theorem1Assumethat（ρ0，u0）∈Hs（R2）×Hs（R2）forsomes>2.Thenthereexistsauniquelocalclassicalsolution（ρ，u）∈C（[0，T）；Hs（R2）×Hs（R2））totheCauchyproblem（1）-（2），forsomeT=T（∥ρ0∥Hs（R2），∥u0∥Hs（R2））.Therestof
AbstractIn this paper， the authors study the local existence of classical solution of the 2D isentropic compressible Eule摘自：毕业论文提纲范文www.7ctime.com
r 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， 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 equations is used to describe the perfectfluids which corresponds to the particular case of Nier-Stokes equations. The Nier-Stokes equations for isentropic compressibleflow in two dimension can be express in the form
rnal force， the viscosity coefficientsλandμsatiyλ> 0，≥
Many results concerning the local existence of equations （3） can be found in [1-4] whenρ0> 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， Schaeffer[12]and McGrath[13]researched the 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 1Assume that （ρ0，u0）∈Hs（R2）×Hs（R2） for some s > 2. Then there exists a unique local classical solution （ρ，u）∈C（[0，T）；Hs（R2）×Hs（R2）） to the Cauchy problem （1）-（2）， for some T = T（∥ρ0∥Hs（R2），∥u0∥Hs（R2））.
The rest of the paper is organized as论文导读：3-253.YCho，HJChoe，HKim.Uniquesolvabilityoftheinitialboundaryvalueproblemorcompressibleviscouluid.JMathPuresAp上一页1234下一页
follows： In Section 2， we state some elementary facts and inequalities which 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 thatρk， ukconverge toρ， u in C（[0，T？]；Hs？1）， C（[0，T？]；Hs？1） as k→+∞， respectively. From the solution of （8）（9）， it is easy to know that （ρ，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 （ρ1，u1） and（ρ2，u2） are both two solutions of the problem （1） （2）， 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.N Itaya. On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscouluids. Kodai Math Sem Rep， 1971， 23： 60-120.
N Itaya. On the initial value problem of the motion of compressible vis源于：大学生论文www.7ctime.com
couluid， especially on the problem of uniqueness. J Math Kyoto Univ， 1976， 16： 413-427.
[3] J Nash. Le probléme de Cauchy pour leséquations diff′erentielles d’unfluid général. Bull Soc Math France， 1962， 90： 487-497.
[4] A Tani. On thefirst initial-boundary value problem of compressible viscouluid motion. Publ Res Inst Math Sci Kyoto Univ， 1971， 13： 193-253.
[5] Y Cho， H J Choe， H Kim. Unique solvability of the initial boundary value problems for compressible viscouluid. J Math Pures Ap论文导读：ions.SciCMath，2010，53（3）：671-686.XDHuang，JLi，ZPXin.Blowupcriterionforviscousbaratropicflowswithvacuumstates.CommMathPhy，2011，301（1）：23-35.YZSun，CWang，ZFZhang.ABeale-Kato-Majdablow-upcriterionfor3-DcompressibleNier-Stokesequations.JMathPures
pl， 2004， 83： 243-275.
[6] Y Cho， H Kim. Strong solutions of the Nier-Stokes equations for isentropic compressiblefluids. J DiffE， 2003， 190： 504-523.
[7] Y Cho， H Kim. On classical solutions of the compressible Nier-Stokes equations with nonnegative initial densities. Manuscripta Math， 2006， 120： 91-129.
[8] D Y 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 Nier-Stokes equations. Sci C Math， 2010， 53（3）： 671-686.
[10] X D Huang， J 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 Nier- Stokes equations. J Math Pures A， 2011， 95（1）： 36-47.
[12] A C Schaeffer. Existence theorem for theflow of an incompressiblefluid 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 C∞solutions of the Euler equations of incompressible perfectfluids. 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 dissipation 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-lin论文导读：earsymmetrichyperbolicsystems.ArchRationMechAnal，1975，58：181-20
AbstractIn this paper， the authors study the local existence of classical solution of the 2D isentropic compressible Eule摘自：毕业论文提纲范文www.7ctime.com
r 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， 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 equations is used to describe the perfectfluids which corresponds to the particular case of Nier-Stokes equations. The Nier-Stokes equations for isentropic compressibleflow in two dimension can be express in the form
rnal force， the viscosity coefficientsλandμsatiyλ> 0，≥
Many results concerning the local existence of equations （3） can be found in [1-4] whenρ0> 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， Schaeffer[12]and McGrath[13]researched the 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 1Assume that （ρ0，u0）∈Hs（R2）×Hs（R2） for some s > 2. Then there exists a unique local classical solution （ρ，u）∈C（[0，T）；Hs（R2）×Hs（R2）） to the Cauchy problem （1）-（2）， for some T = T（∥ρ0∥Hs（R2），∥u0∥Hs（R2））.
The rest of the paper is organized as论文导读：3-253.YCho，HJChoe，HKim.Uniquesolvabilityoftheinitialboundaryvalueproblemorcompressibleviscouluid.JMathPuresAp上一页1234下一页
follows： In Section 2， we state some elementary facts and inequalities which 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 thatρk， ukconverge toρ， u in C（[0，T？]；Hs？1）， C（[0，T？]；Hs？1） as k→+∞， respectively. From the solution of （8）（9）， it is easy to know that （ρ，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 （ρ1，u1） and（ρ2，u2） are both two solutions of the problem （1） （2）， 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.N Itaya. On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscouluids. Kodai Math Sem Rep， 1971， 23： 60-120.
N Itaya. On the initial value problem of the motion of compressible vis源于：大学生论文www.7ctime.com
couluid， especially on the problem of uniqueness. J Math Kyoto Univ， 1976， 16： 413-427.
[3] J Nash. Le probléme de Cauchy pour leséquations diff′erentielles d’unfluid général. Bull Soc Math France， 1962， 90： 487-497.
[4] A Tani. On thefirst initial-boundary value problem of compressible viscouluid motion. Publ Res Inst Math Sci Kyoto Univ， 1971， 13： 193-253.
[5] Y Cho， H J Choe， H Kim. Unique solvability of the initial boundary value problems for compressible viscouluid. J Math Pures Ap论文导读：ions.SciCMath，2010，53（3）：671-686.XDHuang，JLi，ZPXin.Blowupcriterionforviscousbaratropicflowswithvacuumstates.CommMathPhy，2011，301（1）：23-35.YZSun，CWang，ZFZhang.ABeale-Kato-Majdablow-upcriterionfor3-DcompressibleNier-Stokesequations.JMathPures
pl， 2004， 83： 243-275.
[6] Y Cho， H Kim. Strong solutions of the Nier-Stokes equations for isentropic compressiblefluids. J DiffE， 2003， 190： 504-523.
[7] Y Cho， H Kim. On classical solutions of the compressible Nier-Stokes equations with nonnegative initial densities. Manuscripta Math， 2006， 120： 91-129.
[8] D Y 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 Nier-Stokes equations. Sci C Math， 2010， 53（3）： 671-686.
[10] X D Huang， J 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 Nier- Stokes equations. J Math Pures A， 2011， 95（1）： 36-47.
[12] A C Schaeffer. Existence theorem for theflow of an incompressiblefluid 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 C∞solutions of the Euler equations of incompressible perfectfluids. 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 dissipation 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-lin论文导读：earsymmetrichyperbolicsystems.ArchRationMechAnal，1975，58：181-20