On the periodic Zakharov-Kuznetsov equation

Abstract

We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on T 2 . We prove the local well-posedness for given data in H s (T 2 ) whenever s > 5/3. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the R 2 and R × T settings.1991 Mathematics Subject Classification. Primary: 35Q53. Secondary: 35B05.

On short-time bilinear Strichartz estimates and applications to the Shrira equation

“…To conclude the section we compare the linear Strichartz estimate from [19] to the bilinear estimate. For (2)…”

“…For the Zakharov-Kuznetsov equation it is currently unknown whether there is a suitable gauge transform. Using shorttime linear Strichartz estimates, very recently local well-posedness of (2) was proved for s > 5/3 in [19] and in fact, for any period lengths. On R 2 the Zakharov-Kuznetsov equation is known to be semilinear and locally well-posed for s > 1/2 (cf.…”

Periodic Cauchy problem for one two-dimensional generalization of the Benjamin–Ono equation in Sobolev spaces of low regularity

“…The definition of the periodic Sobolev spaces H s (T 2 ) is given in section 2. [14]) in which it suffices to apply the two-dimensional Poisson summation formula, arises from the fact that in our case the symbol e −it sgn(m)(m 2 +n 2 ) is not an smooth function in R 2 m,n .…”

“…Inspired by the works [7] and [14], in this paper we consider the two-dimensional BO equation in the periodic setting. The statement of our result is as follows.…”

“…On the other hand, in order to obtain the energy estimate, contained in Lema 4.3, it is necessary to use a Kato-Ponce commutator estimate in the periodic context (see Lemma 4.1), which was proved in [14].…”