<p>In this paper, we show the shock formation of the solutions to the 3-dimensional (3D) compressible isentropic and irrotational Euler equations with damping for the initial short pulse data, which was first introduced by Christodoulou&#xa0;[<CitationRef CitationID="CR9">9</CitationRef>]. Due to the damping effect, the largeness of the initial data is necessary for the shock formation, and we will work on the class of large data (in the energy sense). Similar to the undamped case, the formation of shocks is characterized by the collapse of the characteristic hypersurfaces and the vanishing of the inverse foliation density function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, at which the first derivatives of the velocity and the density blow up. However, the damping effect changes the asymptotic behavior of the inverse foliation density function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and then shifts the time of shock formation compared with the undamped case. The methods in the paper can also be extended to a class of 3D quasilinear wave equations for the short pulse initial data.</p>

错误:搜索内容不能为空,请输入英文关键词
错误:关键词超出字数限制,请精简
高级检索

Formation of shifted shocks for the 3D compressible Euler equations with damping

  • Zhendong Chen

摘要

In this paper, we show the shock formation of the solutions to the 3-dimensional (3D) compressible isentropic and irrotational Euler equations with damping for the initial short pulse data, which was first introduced by Christodoulou [9]. Due to the damping effect, the largeness of the initial data is necessary for the shock formation, and we will work on the class of large data (in the energy sense). Similar to the undamped case, the formation of shocks is characterized by the collapse of the characteristic hypersurfaces and the vanishing of the inverse foliation density function \(\mu \) μ , at which the first derivatives of the velocity and the density blow up. However, the damping effect changes the asymptotic behavior of the inverse foliation density function \(\mu \) μ and then shifts the time of shock formation compared with the undamped case. The methods in the paper can also be extended to a class of 3D quasilinear wave equations for the short pulse initial data.