On the Cauchy problem of dispersive Burgers type equations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912639890751488 |
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| author | Said, Ayman Rimah |
| author_facet | Said, Ayman Rimah |
| contents | We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[,$$ which contains the main non linear "worst interaction" terms, that is low-high interaction terms, of the usual weakly dispersive Burgers type equation: \[ \partial_t u+u\partial_x u+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[, \] with $u_0 \in H^s({\mathbb D})$, where ${\mathbb D}={\mathbb T} \text{ or } {\mathbb R}$. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [42], we prove a new a priori estimate in $H^s({\mathbb D})$ under the control of $\left\Vert D^{2-α}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$, improving upon the usual hyperbolic control $\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured in [31]. For $α\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{α-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ α\in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-α)^+}_*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_03588 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Cauchy problem of dispersive Burgers type equations Said, Ayman Rimah Analysis of PDEs We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[,$$ which contains the main non linear "worst interaction" terms, that is low-high interaction terms, of the usual weakly dispersive Burgers type equation: \[ \partial_t u+u\partial_x u+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[, \] with $u_0 \in H^s({\mathbb D})$, where ${\mathbb D}={\mathbb T} \text{ or } {\mathbb R}$. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [42], we prove a new a priori estimate in $H^s({\mathbb D})$ under the control of $\left\Vert D^{2-α}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$, improving upon the usual hyperbolic control $\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured in [31]. For $α\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{α-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ α\in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-α)^+}_*$. |
| title | On the Cauchy problem of dispersive Burgers type equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2103.03588 |