具有一般短脉冲数据的三维定常超声速流动的激波形成
Shock formation for 3D steady supersonic flows with general short pulse data
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中文总结 AI 辅助
本文针对多方气体三维定常超声速位势方程，通过构造合适未知量消除相容性条件并推导加权能量估计，证明满足条件的短脉冲超声速边界数据会在有限距离内形成激波，方法可推广至欧拉方程相关问题。
中文摘要 AI 辅助
本文研究多方气体的三维定常超声速位势方程的激波形成问题。该位势方程由二阶拟线性波动方程描述：$\boldsymbol{\textstyle\bigsum_{i=1}^{3}\big[(\boldsymbol{\textstyle\boldsymbol{\nabla}_i\boldsymbol{\nabla}\boldsymbol{\nabla}})^2 - c^2(\rho)\big]\boldsymbol{\nabla}_i^2\boldsymbol{\nabla}\boldsymbol{\nabla}} + 2\boldsymbol{\textstyle\bigsum_{1\boldsymbol{\nabla}i<j\boldsymbol{\nabla}3}\boldsymbol{\nabla}_i\boldsymbol{\nabla}\boldsymbol{\nabla}\boldsymbol{\nabla}_j\boldsymbol{\nabla}\boldsymbol{\nabla}\boldsymbol{\nabla}_{ij}^2\boldsymbol{\nabla}\boldsymbol{\nabla} = 0$，其中$x=(x^1,x^2,x^3)$，$(\boldsymbol{\nabla}_1,\boldsymbol{\nabla}_2,\boldsymbol{\nabla}_3)=(\boldsymbol{\nabla}_{x^1},\boldsymbol{\nabla}_{x^2},\boldsymbol{\nabla}_{x^3})$，$c(\rho)=\boldsymbol{\nabla}p'(\rho)$为声速，$p(\rho)=A\rho^\boldsymbol{\nabla}$（$\boldsymbol{\nabla}>1$），且$\boldsymbol{\nabla}_3\boldsymbol{\nabla}\boldsymbol{\nabla} > c(\rho)$。对于短脉冲边界数据$\boldsymbol{\nabla}\boldsymbol{\nabla}|_{x^3=0} = \boldsymbol{\nabla}^\boldsymbol{\nabla}\boldsymbol{\nabla}_0\big(\frac{r-1}{\boldsymbol{\nabla}},\boldsymbol{\nabla}\big)$和$\boldsymbol{\nabla}_3\boldsymbol{\nabla}\boldsymbol{\nabla}|_{x^3=0}=q_0+\boldsymbol{\nabla}^{\boldsymbol{\nabla}-1}\boldsymbol{\nabla}_1\big(\frac{r-1}{\boldsymbol{\nabla}},\boldsymbol{\nabla}\big)$，其中$r=\boldsymbol{\nabla}(x^1)^2+(x^2)^2$，$\boldsymbol{\nabla}=\big(\frac{x^1}{r},\frac{x^2}{r}\big)\boldsymbol{\nabla}\boldsymbol{\nabla}$，$1<\boldsymbol{\nabla}<2$且小$\boldsymbol{\nabla}>0$，研究表明只要边界数据为超声速且满足$(\boldsymbol{\nabla}_0,\boldsymbol{\nabla}_1)\not\boldsymbol{\nabla}0$，就会在有限的$x^3$距离内形成激波，这与超声速多方气体的强压缩会产生激波的物理现象一致。本文的主要成果之一是找到一个合适的未知量，从而消除了短脉冲初数据上先前施加的相容性条件，同时推导了现有文献中所需的加权能量估计。预计此处的方法将应用于研究多方气体的三维定常超声速欧拉方程的一般短脉冲初数据的激波形成问题。
英文摘要
This paper concerns the shock formation problem for the 3D steady supersonic potential equation of polytropic gases. The potential equation is described by a second order quasilinear wave equation $\displaystyle\sum_{i=1}^{3}\big[(\partial_iΦ)^2 - c^2(ρ)\big]\partial_i^2Φ+ 2\displaystyle\sum_{1\le i<j\le 3}\partial_iΦ\partial_jΦ\partial_{ij}^2Φ= 0$, where $x = (x^1,x^2,x^3)$, $(\partial_1,\partial_2,\partial_3)=(\partial_{x^1},\partial_{x^2},\partial_{x^3})$, $c(ρ) = \sqrt{p'(ρ)}$ is the sonic speed with $p(ρ)=Aρ^γ$ ($γ>1$), and $\partial_3Φ> c(ρ)$. For the short pulse boundary data $Φ|_{x^3=0} = δ^νΦ_0\big(\frac{r-1}δ,ω\big)$ and $\partial_3Φ|_{x^3=0}=q_0+δ^{ν-1}Φ_1\big(\frac{r-1}δ,ω\big)$ with $r=\sqrt{(x^1)^2+(x^2)^2}$, $ω=\big(\frac{x^1}{r},\frac{x^2}{r}\big)\in\mathbb{S}$, $1<ν<2$ and small $δ>0$, it is shown that a shock will be formed in a finite $x^3$-distance as long as the boundary data are supersonic and satisfy $(Φ_0,Φ_1)\not\equiv 0$. This coincides with physical phenomenon that strong compression of supersonic polytropic gases yields shocks. One of our main ingredients is to find a good unknown so that the previously imposed compatibility conditions on the short pulse initial data are removed as well as the required weighted energy estimates in the existing literatures are derived. It is expected that the method here will be applied to study the shock formation problem with general short pulse initial data for the 3D steady supersonic Euler equations of polytropic gases.
