Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation
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2025
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| author | Molinet, Luc Pilod, Didier Vento, Stéphane |
| author_facet | Molinet, Luc Pilod, Didier Vento, Stéphane |
| contents | We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \partial_tu-D_x^α\partial_xu=\partial_x(u^2) \, ,$$ and prove that it is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_α$, where \begin{equation*} s_α=\begin{cases}
1-\frac{3α}4 & \text{for} \quad \frac23 \le α\le 1;
\frac 32(1-α) & \text{for} \quad \frac13 \le α\le \frac23;
\frac 32-\fracα{1-α} & \text{for} \quad 0 < α\le \frac13 .
\end{cases} \end{equation*} The uniqueness is unconditional in $H^s(\mathbb K)$ for $s>\max\{\frac12,s_α\}$. Moreover, we obtain \emph{a priori} estimates for the solutions at the lower regularity threshold $s>\widetilde{s}_α$ where \begin{equation*} \widetilde{s}_α=\begin{cases}
\frac 12-\frac α4 & \text{for} \quad \frac23 \le α\le 1;
1-α& \text{for} \quad \frac12 \le α\le \frac23;
\frac 32-\fracα{1-α} & \text{for} \quad 0 < α\le \frac12 .
\end{cases} \end{equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\mathbb K)$ for $s>s_α$ when $α>\frac23$, and in the energy space $H^{\fracα2}(\mathbb K)$ when $α>\frac45$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation Molinet, Luc Pilod, Didier Vento, Stéphane Analysis of PDEs 35A01, 35A02, 35B45, 35E15, 35Q53 We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \partial_tu-D_x^α\partial_xu=\partial_x(u^2) \, ,$$ and prove that it is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_α$, where \begin{equation*} s_α=\begin{cases} 1-\frac{3α}4 & \text{for} \quad \frac23 \le α\le 1; \frac 32(1-α) & \text{for} \quad \frac13 \le α\le \frac23; \frac 32-\fracα{1-α} & \text{for} \quad 0 < α\le \frac13 . \end{cases} \end{equation*} The uniqueness is unconditional in $H^s(\mathbb K)$ for $s>\max\{\frac12,s_α\}$. Moreover, we obtain \emph{a priori} estimates for the solutions at the lower regularity threshold $s>\widetilde{s}_α$ where \begin{equation*} \widetilde{s}_α=\begin{cases} \frac 12-\frac α4 & \text{for} \quad \frac23 \le α\le 1; 1-α& \text{for} \quad \frac12 \le α\le \frac23; \frac 32-\fracα{1-α} & \text{for} \quad 0 < α\le \frac12 . \end{cases} \end{equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\mathbb K)$ for $s>s_α$ when $α>\frac23$, and in the energy space $H^{\fracα2}(\mathbb K)$ when $α>\frac45$. |
| title | Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation |
| topic | Analysis of PDEs 35A01, 35A02, 35B45, 35E15, 35Q53 |
| url | https://arxiv.org/abs/2506.17801 |