On the Cauchy problem for dispersive Burgers type equations

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 introduced previously by the author 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: \(\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x \abs{D}^{\alpha-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ \alpha \in ]2,3[,\) where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-\alpha)^+}_*$.

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