二维等熵可压欧拉方程古典解的存在性(英文)论文
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最后更新时间:2024-02-20
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， 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 follows： In Section 2， we state some elementary facts andinequalities 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.
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 follows： In Section 2， we state some elementary facts andinequalities 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.