Endpoint Sobolev Theory for the Muskat Equation

doi:10.1007/s00220-022-04514-7

	Endpoint Sobolev Theory for the Muskat Equation
	Alazard, Thomas 1; Quoc-Hung Nguyen 2
刊名	COMMUNICATIONS IN MATHEMATICAL PHYSICS
	2022-11-12
页码	60
ISSN号	0010-3616
DOI	10.1007/s00220-022-04514-7
英文摘要	This paper is devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. We prove that the Cauchy problem is well-posed on the endpoint Sobolev space of L-2 functions with three-half derivative in L-2. This result is optimal with respect to the scaling of the equation. One well-known difficulty is that one cannot define a flow map such that the lifespan is bounded from below on bounded subsets of this critical Sobolev space. To overcome this, we estimate the solutions for a norm which depends on the initial data themselves, using the weighted fractional Laplacians introduced in our previous works. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.
资助项目	French National Research Agency (ANR)[ANR-18-CE40-0027] ; Academy ofMathematics and Systems Science, Chinese Academy of Sciences startup fund ; National Natural Science Foundation of China[12050410257] ; National Natural Science Foundation of China[12288201] ; National Key R&D Program of China[2021YFA1000800]
WOS研究方向	Physics
语种	英语
出版者	SPRINGER
WOS记录号	WOS:000882353600003
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/60613]
专题	中国科学院数学与系统科学研究院
通讯作者	Alazard, Thomas
作者单位	1.Univ Paris Saclay, Ctr Borelli UMR9010, ENS Paris Saclay, CNRS, Ave Sci, F-91190 Gif Sur Yvette, France 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Alazard, Thomas,Quoc-Hung Nguyen. Endpoint Sobolev Theory for the Muskat Equation[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS,2022:60.
APA	Alazard, Thomas,&Quoc-Hung Nguyen.(2022).Endpoint Sobolev Theory for the Muskat Equation.COMMUNICATIONS IN MATHEMATICAL PHYSICS,60.
MLA	Alazard, Thomas,et al."Endpoint Sobolev Theory for the Muskat Equation".COMMUNICATIONS IN MATHEMATICAL PHYSICS (2022):60.