On the (generalized) Korteweg-de Vries equation
On the (generalized) Korteweg-de Vries equation
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DOI：
10.1215/s0012-7094-89-05927-9
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发表时间：
1989-12
影响因子：
2.5
通讯作者：
C. Kenig;G. Ponce;L. Vega
中科院分区：
文献类型：
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作者：
C. Kenig;G. Ponce;L. Vega
其中a是关于R to的函数，其值为(0)0，正则性将在后面指定。我们将在经典的Sobolev空间HS(R)中研究这些问题的适定性，并在空间Lff(1A)-S/2LP()中研究它们解的正则性。在适定性中，我们包括存在性、唯一性、持久性(即在时刻[T，T]的解u(T)与初始数据UO属于同一函数空间X，并且描述了X中的连续曲线)，以及映射UO u(T)从X到C([-T，T]：X)的连续性。如果T T(Lluollx)<；，我们称它在X中是局部适定的。在T可以取任意大的情况下，问题在X中是全局适定的。下面我们的主要结果之一是证明了这些方程的解的全局(在空间)光滑化效应。为了解释它，我们首先考虑与UO L2()相关的线性问题(即，(1.2)中的a(.)0)。在这种情况下，解u(T)由酉群{W(T)}_oo给出。因此，u(T)e‘OUO W(T)UO C([：L2()。从1-12][13][19]中的结果可以得出如下R.S.Strichartz类型的结果[17]：
where a is a function on R to with a(0) 0 and regularity to be specified later. We shall study the well-posedness of these problems in the classical Sobolev spaces Hs(R), and the regularity of their solutions in the spaces Lff (1A)-S/2Lp(). In well-posedness we include existence, uniqueness, persistence property (i.e., the solution u(t) at the time [T, T] belongs to the same function space X as does the initial data Uo, and describes a continuous curve in X), and the continuity of the map Uo u(t) from X to C([-T, T]: X). If T T(lluollx)< we call it local well-posed in X. In the case when T can be taken arbitrarily large the problem is globally well-posed in X. One of our main results below is the proof of a global (in space) smoothing effect for solutions of these equations. To explain it we consider first the associated linear problem (i.e., a(.) 0 in (1.2)) with Uo L2(). In this case the solutions u(t) is given by the unitary group {W(t)}_oo. Thus, u(t) e’OUo W(t)Uo C([: L2()). From the results in 1-12] [13] [19] one has the following R. S. Strichartz type of result [17]: