On the Cauchy problem of dispersive Burgers type equations

Abstract

We study the paralinearised weakly dispersive Burgers type equation: $${\partial }_{t}u+T_u{\partial }_{x}u+{\partial }_x|D{|}^{\alpha -1}u=0,\alpha \in ]1,2[,$$ 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 ${\Vert D^{2-\alpha }\left(u^2\right)\Vert }_{L_t^1{L}_x^{\infty }}$, improving upon the usual hyperbolic control ${\Vert {\partial }_{x}u\Vert }_{L_t^1{L}_x^{\infty }}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured in [31]. For $\alpha \in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $${\partial }_t\left[T_{e^{i{T}_{p(u)}}}u\right]+{\partial }_x|D{|}^{\alpha -1}\left[T_{e^{i{T}_{p(u)}}}u\right]=T_{R(u)}u,\alpha \in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L_t^{\infty }{C}_{\ast }^{(2-\alpha {)}^{+}}$.

Updated version after review which closely follows the journal version to appear in Indiana University Mathematics Journal, 2022. arXiv admin note: text overlap with arXiv:2103.03576

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• The wave breaking for Whitham-type equations revisited
• On the flow map of the Benjamin-Ono equation on the torus
• A geometric proof of the Quasi-linearity of the water-waves system
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Cited by in corpus (1)

• Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation