On a higher dimensional version of the Benjamin--Ono equation

On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations

“…In higher dimensions (1) yields a generalization of the Benjamin-Ono equation for a = 1 (cf. [25,26,29]) and for a = 2 the Zakharov-Kuznetsov equation (cf. [36]) is recovered.…”

The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces

“…The results concerning local well-posedness for the (HBO) equation in classical Sobolev spaces H s (R d ) are fundamental in our arguments to extend these conclusions to weighted spaces. In this regard, we recall the following results derived in [14].…”

“…Here we adopt Kato's notion of well-posedness, which consists of existence, uniqueness, persistence property (i.e., if the data u 0 ∈ X a function space, then the corresponding solution u(•) describes a continuous curve in X, u ∈ C([0, T ]; X), T > 0), and continuous dependence of the map data-solution. Regarding the IVP for (HBO), in [14] LWP was deduced for s > 5/3 when d = 2 and for s > (d + 1)/2 when d ≥ 3. In [21], LWP was improved to the range s > 3/2 in the case d = 2.…”

“…Up to our knowledge there are no results concerning global well-posedness (GWP) in the current literature. It is worthwhile to mention that local well-posedness issues have been addressed by compactness methods, since one cannot solve the IVP related to (HBO) by a Picard iterative method implemented on its integral formulation for any initial data in the Sobolev space H s (R d ), d ≥ 2 and s ∈ R. This is a consequence of the results deduced in [14], where it was established that the flow map data-solution u 0 → u for (HBO) is not of class C 2 at the origin from…”

“…(ii) The restrictions on the Sobolev regularity stated in Proposition 1.1 and Theorem 1.1 are imposed from our recent results in [14], which assure that under such considerations the solution u(x, t) satisfies…”