Abstract. Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV-or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state may be accurate on a much longer time scale. The initial-value problem for the class of equations that emerges from our derivation is then considered. A local wellposedness theory is straightforwardly established by way of a contraction mapping argument.A subclass of these equations possess a special Hamiltonian structure that implies the local theory can be continued indefinitely.
In this paper we study the theory of operators on complex Hilbert spaces, which achieve the norm in the unit sphere. We prove important results concerning the characterization of the AN operators, see Definition 1.2. The class of the AN operators contains the algebra of the compact ones.
Mathematics Subject Classification (2010). Primary 47B07;Secondary 47A58, 47B65, 47A12, 47A15.
Abstract. Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces H s (R)× H s (R) for 3/4 < s ≤ 1. We introduce some Bourgain-type spaces X a s,b for a = 0, s, b ∈ R to obtain local well-posedness for the Gear-Grimshaw system in H s (R) × H s (R) for s > −3/4, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces X
MSC: 35A07; 35Q53
We prove that, if a sufficiently smooth solution u to the initial value problem associated with the equationis supported in a half line at two different instants of time then u ≡ 0. To prove this result we derive a new Carleman type estimate by extending the method introduced by Kenig et al. in [Ann. Inst.
In this paper we study the theory of operators on complex Hilbert spaces, which attain their minima in the unit sphere. We prove some important results concerning the characterization of the N * , and also AN * operators, see respectively Definition 1.1 and Definition 1.3. The injective property plays an important role in these operators, and shall be established by these classes.
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