Unconditional Uniqueness of 5th Order KP Equations

TL;DR

This paper proves the unconditional uniqueness of solutions for 5th order KP equations on the real line by combining energy estimates, short-time $X^{s,b}$ methods, and multilinear interpolation to achieve near-sharp regularity.

Contribution

It adapts and extends energy estimate techniques with short-time $X^{s,b}$ methods and multilinear interpolation to establish unconditional uniqueness for 5th order KP equations.

Findings

Unconditional uniqueness holds for solutions with regularity close to $L^2$.

Application of short-time $X^{s,b}$ methods improves decay estimates.

Multilinear interpolation enables access to $L^4$ Strichartz estimates.

Abstract

In this paper we study the $5$th Order Kadomstev-Petviashvili (KP) equations posed on the real line. In particular we adapt the energy estimate argument from Guo-Molinet (arXiv:2404.12364v1 [math.AP]) to conclude unconditional uniqueness of the solution to data map for $5$th order KP type equations. Applying short-time $X^{s,b}$ methods to improve classical energy estimates provides more than sufficient decay when considering estimates on the interior of the time interval $[0,T]$. The issue is how we deal with the boundary. By abusing symmetry we can apply multilinear interpolation to gain access to $L^4$ Strichartz estimates, which provide improved derivative gain. When taken together, the regularity of our resultant function space can be arbitrarily close to $L^2$, which in the context of unconditional uniqueness results is almost sharp.