On decay properties for solutions of the Zakharov–Kuznetsov equation

Abstract

This work mainly focuses on spatial decay properties of solutions to the Zakharov–Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition $u_0$ verifies ${〈\sigma \cdot x〉}^r{u}_0\in L^2\left(\left(\sigma \cdot x\ge \kappa \right)\right),$ for some $r\in N$, $\kappa \in R$, being $\sigma$ be a suitable non-null vector in the Euclidean space, then the corresponding solution $u\left(t\right)$ generated from this initial condition verifies ${〈\sigma \cdot x〉}^{r}u\left(t\right)\in L^2\left(\left(\sigma \cdot x>\kappa -\nu t\right)\right)$, for any $\nu >0$. Additionally, depending on the magnitude of the weight $r$, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power $r>0$ not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data $u_0$ has a decay of exponential type on a particular half space, that is, $e^{b\sigma \cdot x}{u}_0\in L^2\left(\left(\sigma \cdot x\ge \kappa \right)\right),$ then the corresponding solution satisfies $e^{b\sigma \cdot x}u\left(t\right)\in H^p\left(\left(\sigma \cdot x>\kappa -t\right)\right),$ for all $p\in N$, and time $t\ge \delta ,$ where $\delta >0$. To our knowledge, this is the first study of such property. As a further consequence, we also obtain well-posedness results in anisotropic weighted Sobolev spaces in arbitrary dimensions.

Finally, as a by-product of the techniques considered here, we show that our results are also valid for solutions of the Korteweg–de Vries equation.