On the Cauchy problem of dispersive Burgers type equations

Abstract

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ which contains the main non linear "worst interaction" terms, that is low-high interaction terms, of the usual weakly dispersive Burgers type equation: \[ \partial_t u+u\partial_x u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[, \] 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 $\left\Vert D^{2-\alpha}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$, 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 $\alpha\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{\alpha-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ \alpha \in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-\alpha)^+}_*$.