非线性KdV方程相关半群的Gevrey类正则性研究
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更新于2024-09-04
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本文主要探讨了与一类非线性Korteweg-de Vries (KdV) 方程相关的线性半群在Gevrey类中的正则性问题。Gevrey正则性是泛函分析中衡量函数或分布族在微分方程解中的行为的一个重要概念，它描述了解的光滑程度超越了一般的解析性质。KdV方程是一个广泛研究的非线性偏微分方程，常用于描述浅水波的传播，其线性化后涉及到的线性算子在许多物理问题中扮演着关键角色。
储继迅、Jean-Michel Coron、尚培培和唐姝霞这四位作者合作，他们的研究关注的是这种线性半群是否具有解析性。通过对线性算子预解式进行深入分析，他们发现该线性算子生成的半群并不属于通常意义上的解析半群，而是属于δ-Gevrey类。Gevrey类是一种比解析函数更广泛的函数类，其定义特点是函数的高阶导数的增长速度比一般解析函数更快，这里δ值介于3/2和无穷大之间。
预解式是求解线性微分方程时常用的一种工具，通过估计预解式的性质，可以揭示线性算子的行为特性。在本文中，作者们通过严格的数学推理和估计，得出了关于非线性KdV方程线性化的具体结论，这对于理解和控制这类非线性系统的行为具有重要意义。
此外，文章还涉及到了应用数学的范畴，强调了KdV方程及其线性化在实际问题中的应用，如浅水波的数值模拟、波动理论等。关键词包括应用数学、KdV方程、预解式估计、解析半群和Gevrey类，这些都是研究者在探索这一领域时会参考的关键术语。
这篇首发论文提供了一个重要的理论结果，即对于非线性KdV方程的线性半群，其Gevrey正则性存在限制，这不仅深化了对非线性微分方程理论的理解，也为相关领域的数值计算和模型建立提供了理论支持。
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the linear operator A. To be more precise, it is proved in [1] that when L ∈ N, (0.2) admits a
family of non-trivial solutions of the form y
0
(x)e
λt
for some λ ∈ iR, where y
0
(x) satisfies
λy
0
(x) + y
′
0
(x) + y
′′′
0
(x) = 0,
y
0
(0) = y
0
(L) = y
′
0
(0) = y
′
0
(L) = 0.
In this case, the origin of (0.2) is not asymptotically stable, and thus the linearization analysis
fails. Still, as proposed in [3, 4], it remains very interesting to study the asymptotic properties
of the nonlinear system (0.1) in this critical case.
In [5], we prove the existence and smoothness of the center manifold of (0.1) when L = 2kπ,
(i.e., taking j = l = k in N), where k is a positive integer such that (see [3, Theorem 8.1 and
Remark 8.2])
j
2
+ l
2
+ jl = 3k
2
and j, l ∈ N
∗
⇒ (j = l = k) . (0.4)
In this case, the center manifold is of dimension 1. By analyzing the reduced equation on
the center manifold, we show that the nonlinear system (0.1) is asymptotically stable around
the origin with a polynomial decay rate. Using the same method, we prove in [6] the local
asymptotic stability for a special critical length L = 2π
7/3 corresponding to j = 1 and l = 2
in the following critical case [7]
N
′
:=
2π
j
2
+ l
2
+ jl
3
; j, l ∈ N
∗
satisfying j > l and
j
2
+ jl + l
2
= m
2
+ mn + n
2
, ∀m, n ∈ N
∗
\{j}
, (0.5)
where the center manifold is of dimension 2.
While studying the existence of the center manifold, it is noticed that the linear operator A
does not generate an analytic semigroup, but a semigroup of Gevrey class. The Gevrey class of
semigroups have a behavior somewhat “between” that of differentiable semigroups and analytic
semigroups. For the convenience of readers, we first give the definition of Gevrey class δ > 1
(see [8, 9]).
Definition 0.1. Let T (t) be a strongly continuous semigroup on a Banach space X and let
δ > 1. We say that T (t) is of Gevrey class δ for t > t
0
if T (t) is infinitely differentiable for
t ∈ (t
0
, ∞) and, for every compact subset K ⊂ (t
0
, ∞) and each θ > 0, there exists a constant
C = C(θ, K) such that
∥T
(n)
(t)∥ 6 Cθ
n
(n!)
δ
, ∀t ∈ K and n = 1, 2, ··· .
When δ = 1, T (t) is analytic [10].
With the definition of Gevrey class δ, we give the main result of this article.
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