On the Cauchy problem of dispersive Burgers type equations

Abstract

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+\partial_x [T_u u]-T_{\frac{\partial_x u}{2}}u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ which contains the main non linear "worst interaction" terms, i.e 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 introduce for the study of the flow map regularity of Gravity-Capillary equation, we prove a new a priori estimate in $H^s(\mathbb D)$ under the control of $\left\Vert(1+\left\Vert u\right\Vert_{L^\infty_x})\left\Vert u \right\Vert_{W^{2-\alpha,\infty}_x}\right\Vert_{L^1_t}$, 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 by J. C. Saut and C. Klein in their numerical study of the dispersive Burgers equation. 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: $$\partial_tu+ \partial_x |D|^{\alpha-1}u=R_\infty(u),\ \alpha \in ]2,3[,$$ where $R_\infty$ is a regularizing operator under the control of $\left\Vert u\right\Vert_{L^\infty_t C^{2-\alpha}_*}$