Well-Posedness of the Initial Value Problem for the Korteweg-de Vries Equation

Abstract

This paper is mainly concerned with the initial value problem (IVP) for the Korteweg-de Vries (KdV) equation { 8tu + 8;u + u8 x u = 0, (1.1) u(x, 0) = uo(x). X,tElR, The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global well-posedness of the IVP (1.1) in classical Sobolev spaces H\lR). We shall say that the IVP (1.1) is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow. It was established in the works of Bona and Smith [3], Bona and Scott [2], Saut and Temam [30], and Kato [15] that the IVP (1.1) is locally (resp. globally) well-posed in H S with s > 3/2 (resp. s ~ 2). Roughly speaking, global well-posedness in H S depends on the available local theory and on the conservation laws satisfied by solutions of (1.1), namely: ~l(U) = ! udx, ~2(U) = ! u 2 dx , ~3(U) = ! ((8 x U)2-~ u 3

Decay of Small Solutions for the Zakharov-Kuznetsov Equation posed on a half-strip

“…More popular are Korteweg-de Vries and Schrödinger equations. The theory of the Cauchy problem for them nowadays is well developed and presented in papers of Bona and his colleagues [1], Kruzhkov and Faminskii [11], Kato [9], Bourgain [3], Saut [21], Temam [23], Ponce and his colleages [10,17], etc. Last years appeared papers on initial boundary value problems for dispersive equations in bounded and non-bounded domains.…”

Odd-order quasilinear evolution equations posed on a bounded interval

“…In fact, consider an initial value problem in a strip Π T = (0, T ) × R for (1.5) with the initial data (1.2). This problem was studied in [23]. In particular, if u 0 ∈ H (2l+1)s (R), then for any T > 0 there exists a solution of (1.5), (1.2), S(t, x; u 0 ), given by the formula…”

“…Therefore, precise mathematical analysis of boundary value problems in bounded domains for general dispersive equations is welcome and attracts attention of specialists in the area of dispersive equations, especially KdV and BBM equations, [3,5,6,7,8,9,12,15,16,17,18,21,25,26,27,28,31,37,38]. Cauchy problem for dispersive equations of high orders was successfully explored by various authors, [2,10,11,15,23,33,36]. On the other hand, we know few published results on initialboundary value problems posed on a finite interval for general nonlinear odd-order dispersive equations, such as the Kawahara equation, see [13,14,29,20].…”

Local well-posedness for the periodic higher order KdV type equations

“…More precisely, the data-to-solution map is not uniformly continuous on H s (T) for −2 < s < −1/2. LWP of the non-periodic KdV equation also was studied by many people before Bourgain's work [1,7,11,13,14] and after Bourgain's work [5,10,15,[17][18][19]21,24]. Next, we introduce some known results for the fifth order KdV type equations ∂ t u + α∂ 5 x u + β∂ 3 x u + ∂ x (u 2 ) = 0 (1.2) and…”