Blow-up solutions of the "bad" Boussinesq equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911882423566336 |
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| author | Charlier, Christophe |
| author_facet | Charlier, Christophe |
| contents | We study blow-up solutions of the ``bad" Boussinesq equation, and prove that a wide range of asymptotic scenarios can happen. For example, for each $T>0$, $x_{0}\in \mathbb{R}$ and $δ\in (0,1)$, we prove that there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that $|u(x,t)| \leq C \frac{1+x^{2}}{(x-x_{0})^{2}}$ and $u(x_{0},t)\asymp (T-t)^{-δ}$ as $t\to T$.
We also prove that for any $q\in \mathbb{N}$, $T>0$, $x_{0}\in \mathbb{R}$, $δ\in (0,\frac{1}{2})$, there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that (i) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)|\leq C$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}\leq q$, (ii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)| \leq C \frac{1+|x|}{|x-x_{0}|}$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$, (iii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x_{0},t)| \asymp (T-t)^{-δ}$ as $t\to T$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$.
In particular, when $q=0$, this result establishes the existence of wave-breaking solutions, i.e. solutions that remain bounded but whose $x$-derivative blows up in finite time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12210 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-up solutions of the "bad" Boussinesq equation Charlier, Christophe Analysis of PDEs We study blow-up solutions of the ``bad" Boussinesq equation, and prove that a wide range of asymptotic scenarios can happen. For example, for each $T>0$, $x_{0}\in \mathbb{R}$ and $δ\in (0,1)$, we prove that there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that $|u(x,t)| \leq C \frac{1+x^{2}}{(x-x_{0})^{2}}$ and $u(x_{0},t)\asymp (T-t)^{-δ}$ as $t\to T$. We also prove that for any $q\in \mathbb{N}$, $T>0$, $x_{0}\in \mathbb{R}$, $δ\in (0,\frac{1}{2})$, there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that (i) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)|\leq C$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}\leq q$, (ii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)| \leq C \frac{1+|x|}{|x-x_{0}|}$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$, (iii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x_{0},t)| \asymp (T-t)^{-δ}$ as $t\to T$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$. In particular, when $q=0$, this result establishes the existence of wave-breaking solutions, i.e. solutions that remain bounded but whose $x$-derivative blows up in finite time. |
| title | Blow-up solutions of the "bad" Boussinesq equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.12210 |