Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes

Abstract

Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations 15 (2002) 1325-1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of the strip in time.

New lower bounds on the radius of spatial analyticity for the KdV equation

“…Grujić and Kalisch [10] showed that, if the datum u 0 belongs to G σ0 for some σ 0 > 0, then the KdV equation (1.1) has a unique solution u ∈ C[−T, T ; G σ0 ] with a lifespan T depending on u 0 G σ 0 . Similar results for the periodic KdV equation are proved by Hannah, Himonas and Petronilho [23,24] and Li [20]. The work by Grujić and Kalisch [10] improved the earlier results of Hayashi [21,22], where the analyticity radius σ(t) of local solution may depend on t. The local well-posedness in [9,10,23,24,20] shows that, for short times, the KdV equation persists the uniform radius of spatial analyticity as time progresses.…”

“…Similar results for the periodic KdV equation are proved by Hannah, Himonas and Petronilho [23,24] and Li [20]. The work by Grujić and Kalisch [10] improved the earlier results of Hayashi [21,22], where the analyticity radius σ(t) of local solution may depend on t. The local well-posedness in [9,10,23,24,20] shows that, for short times, the KdV equation persists the uniform radius of spatial analyticity as time progresses. Now we turn to the global well-posedness.…”

Lower Bounds on the Radius of Spatial Analyticity for the KdV Equation

“…This question has received some attention in the case of the KdV equation and its generalizations. For short times, it is known that the radius of analyticity remains at least as large as the initial radius; see Grujić and Kalisch [6] for the non-periodic case, and also Li [15], Himonas and Petronilho [8], and Hannah, Himonas and Petronilho [7] for the periodic case. For the global problem, the non-periodic case was studied by Bona, Grujić and Kalisch in [2], where it was shown that the radius of analyticity for the KdV equation can decay no faster than t −12 as t → ∞ (see Theorem 4 and Corollary 2 in [2]).…”

On the Radius of Spatial Analyticity for the Quartic Generalized KdV Equation

“…In this paper we use the idea introduced in [17] (see also [16]) to improve this result significantly showing σ(t ) can decay no faster than t −2 as t → ∞. For studies on related issues for nonlinear partial differential equations see for instance [1,3,9,10,11,12,14,15]. The Gevrey space, denoted G σ,s = G σ,s (R), is a suitable space to study analyticity of solution.…”