On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping
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arXiv
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| Natura: | Preprint |
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2016
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| author | Hou, Fei Yin, Huicheng |
| author_facet | Hou, Fei Yin, Huicheng |
| contents | In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*}
\partial_tρ+\operatorname{div}(ρu)=0, \quad
\partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_d\right)=-α(t)ρu, \quad
ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), \end{equation*} where $x=(x_1, \cdots, x_d)\in\Bbb R^d$ $(d=2,3)$, the frictional coefficient is $α(t)=\fracμ{(1+t)^λ}$ with $λ\ge0$ and $μ>0$, $\barρ>0$ is a constant, $ρ_0,u_0 \in C_0^\infty(\Bbb R^d)$, $(ρ_0,u_0)\not\equiv 0$, $ρ(0,x)>0$, and $\varepsilon>0$ is sufficiently small. One can totally divide the range of $λ\ge0$ and $μ>0$ into the following four cases:
Case 1: $0\leλ<1$, $μ>0$ for $d=2,3$;
Case 2: $λ=1$, $μ>3-d$ for $d=2,3$;
Case 3: $λ=1$, $μ\le 3-d$ for $d=2$;
Case 4: $λ>1$, $μ>0$ for $d=2,3$.
\noindent We show that there exists a global $C^{\infty}-$smooth solution $(ρ, u)$ in Case 1, and Case 2 with $\operatorname{curl} u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(ρ, u)$ blows up in finite time. Therefore, $λ=1$ and $μ=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(ρ, u)$ in $d-$dimensional compressible Euler equations with time-depending damping. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1606_08935 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping Hou, Fei Yin, Huicheng Analysis of PDEs 35L70, 35L65, 35L67, 76N15 In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*} \partial_tρ+\operatorname{div}(ρu)=0, \quad \partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_d\right)=-α(t)ρu, \quad ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), \end{equation*} where $x=(x_1, \cdots, x_d)\in\Bbb R^d$ $(d=2,3)$, the frictional coefficient is $α(t)=\fracμ{(1+t)^λ}$ with $λ\ge0$ and $μ>0$, $\barρ>0$ is a constant, $ρ_0,u_0 \in C_0^\infty(\Bbb R^d)$, $(ρ_0,u_0)\not\equiv 0$, $ρ(0,x)>0$, and $\varepsilon>0$ is sufficiently small. One can totally divide the range of $λ\ge0$ and $μ>0$ into the following four cases: Case 1: $0\leλ<1$, $μ>0$ for $d=2,3$; Case 2: $λ=1$, $μ>3-d$ for $d=2,3$; Case 3: $λ=1$, $μ\le 3-d$ for $d=2$; Case 4: $λ>1$, $μ>0$ for $d=2,3$. \noindent We show that there exists a global $C^{\infty}-$smooth solution $(ρ, u)$ in Case 1, and Case 2 with $\operatorname{curl} u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(ρ, u)$ blows up in finite time. Therefore, $λ=1$ and $μ=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(ρ, u)$ in $d-$dimensional compressible Euler equations with time-depending damping. |
| title | On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping |
| topic | Analysis of PDEs 35L70, 35L65, 35L67, 76N15 |
| url | https://arxiv.org/abs/1606.08935 |