2021/03/05 by Ayman Rimah Said, Said, Ayman Rimah · 1 citation
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations #Mathematical Analysis and Transform Methods
paper · doi:10.48550/arxiv.2103.03588
We study the paralinearised weakly dispersive Burgers type equation: ∂t u+Tu ∂xu+∂x |D|α-1u=0, α∈ ]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: ∂t u+u∂x u+∂x |D|α-1u=0, α∈ ]1,2[, with u0 ∈ Hs(\mathbb D), where \mathbb D=\mathbb T or \mathbb R. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [42], we prove a new a priori estimate in Hs(\mathbb D) under the control of \Vert D2-α(u2)\VertL1tL∞x, improving upon the usual hyperbolic control \Vert ∂x u\VertL1tL^∞x. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured in [31]. For α∈ ]2,3[ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: ∂t [T_e^iTp(u)u]+ ∂x |D|α-1[T_e^iTp(u)u]=TR(u)u, α∈ ]2,3[, where Tp(u) and TR(u) are paradifferential operators of order 0 defined for u∈ L^∞t C(2-α)+_*.