On the Cauchy problem of dispersive Burgers type equations
Ayman Rimah Said
TL;DR
This work investigates the Cauchy problem for paralinearised weakly dispersive Burgers-type equations $\partial_t u+T_u\partial_xu+\partial_x|D|^{\alpha-1}u=0$ with $\alpha\in(1,3)$, introducing a paradifferential gauge transform to tame the leading low–high frequency interactions. The authors develop a BCH-type calculus for hyperbolic paradifferential flows, construct an implicit symbol $p$ to implement a gauge $A=e^{iT_p}$, and derive a robust $H^s$ energy framework. For $1<\alpha<2$ they obtain a new a priori estimate $\|u(t)\|_{H^s} \le e^{C\|D^{2-\alpha}(u^2)\|_{L^1_tL^{\infty}_x}}\|u_0\|_{H^s}$ under a smallness assumption, improving the usual $\|\partial_x u\|_{L^1_tL^{\infty}_x}$ control and precluding wave breaking. In the higher-dispersion range $2<\alpha<3$, the gauge transform yields a complete conjugation to a semi-linear form $\partial_t [T_{e^{iT_p}}u]+\partial_x|D|^{\alpha-1}[T_{e^{iT_p}}u]=T_{R(u)}u$, with a residual of order $0$, enabling local well-posedness in $H^s$ for $s>\frac{5}{2}-\alpha$. These results extend gauge-transform techniques to non-integrable fractional-dispersion Burgers equations and provide a unified framework bridging hyperbolic and dispersive regimes.
Abstract
We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[,$$ 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 $\left\Vert D^{2-α}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$, 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 $α\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{α-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ α\in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-α)^+}_*$.