Initial boundary value problem for Korteweg–de Vries equation: a review and open problems

Abstract

In the last 40 years the study of initial boundary value problem for the Korteweg-de Vries equation has had the attention of researchers from various research fields. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view. 2010 Mathematics Subject Classification. 35Q53, 35Q35, 53C35. Key words and phrases. KdV equation, Well-posedness, Non-homogeneous boundary value problem, Boundary integral operators, Initial boundary value problem.1 The interested readers are referred to [37,49] for history and origins of the Korteweg-de Vries equation.

Global control aspects for long waves in nonlinear dispersive media

“…With this information in hand, choose the two feedback controls 11), to transform this system in a resulting closed-loop system reads as follows…”

“…From the historical origins of the KdV equation involving the behavior of water waves in a shallow channel [5,11,29,21], it is natural to think of I 1 and I 2 as expressing conservation of volume (or mass) and energy, respectively. The Cauchy problem for equation (1.9) has been intensively studied for many years (see [4,18,19] and the references therein).…”

Characterization of Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey Spaces

“…The Fourier-analytic approach on Herz-type Besov-Morrey and Triebel-Lizorkin-Morrey spaces facilitates the studying of mild solutions of nonlinear PDEs, where the heat semi-group operator and the Lerray projection are intensively engaged. For example, a mathematical model of waves on shallow water surfaces described by Korteweg-de Vries equation [17], Keller-Segel System [18] presents a cellular chemotaxis model, or Fokker-Planck equations [19] demonstrates models of anomalous diffusion processes. The developing of atomic, molecular and wavelet decompositions can advance the studying of Kα p,q Ṅ s µ,r and Kα…”

Estimates of Mild Solutions of Navier–Stokes Equations in Weak Herz-Type Besov–Morrey Spaces

“…The properties of Herz-type Besov-Morrey spaces, such as the interpolations in Theorem 3 and the inequalities in Lemma 1, can be also used to study other nonlinear PDEs. For example, a mathematical model of waves on shallow water surfaces described by Korteweg-de Vries equation [22]; the Keller-Segel system [23] presents a cellular chemotaxis model; and Fokker-Planck equations [24] demonstrate models of anomalous diffusion processes. Developing atomic decomposition, oscillations, real and complex interpolations can advance the study of the W Kα p,q Ṅ s µ,r spaces, especially observing them not only with the Fourier approach ( [25]), but by the finite difference approach, in the same fashion of Besov spaces in [26,27].…”