On the Cauchy problem of dispersive Burgers type equations

Updated version after review which closely follows the journal version to appear in Indiana University Mathematics Journal, 2022We 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 $u\in L^\infty_t C^{(2-\alpha)^+}_*$