The Helmholtz decomposition of vector fields for two-dimensional exterior Lipschitz domains

Keiichi Watanabe; Keiichi Watanabe

doi:10.3934/math.2024870

AIMS Mathematics

2024, Volume 9, Issue 7: 17886-17900. doi: 10.3934/math.2024870

Previous Article Next Article

Research article Special Issues

The Helmholtz decomposition of vector fields for two-dimensional exterior Lipschitz domains

Keiichi Watanabe ^,

School of General and Management Studies, Suwa University of Science, 5000-1, Toyohira, Chino, Nagano 391-0292, Japan

Received: 27 March 2024 Revised: 08 May 2024 Accepted: 21 May 2024 Published: 24 May 2024
MSC : 35J57, 35Q35

Let $ \Omega $ be an exterior Lipschitz domain in $ \mathbb{R}^2 $. It is proved that the Helmholtz decomposition of the vector fields in $ L_p (\Omega; \mathbb{R}^2) $ exists if $ p $ satisfies $ \lvert1/ p - 1/ 2 \rvert < 1/ 4+ \varepsilon $ with some constant $ \varepsilon = \varepsilon (\Omega) \in (0, 1/ 4] $, where it is allowed to take $ \varepsilon = 1/ 4 $ if $ \partial \Omega \in C^1 $.
- exterior Lipschitz domains,
- Helmholtz decomposition
Citation: Keiichi Watanabe. The Helmholtz decomposition of vector fields for two-dimensional exterior Lipschitz domains[J]. AIMS Mathematics, 2024, 9(7): 17886-17900. doi: 10.3934/math.2024870

Related Papers:

Abstract

Let $ \Omega $ be an exterior Lipschitz domain in $ \mathbb{R}^2 $. It is proved that the Helmholtz decomposition of the vector fields in $ L_p (\Omega; \mathbb{R}^2) $ exists if $ p $ satisfies $ \lvert1/ p - 1/ 2 \rvert < 1/ 4+ \varepsilon $ with some constant $ \varepsilon = \varepsilon (\Omega) \in (0, 1/ 4] $, where it is allowed to take $ \varepsilon = 1/ 4 $ if $ \partial \Omega \in C^1 $.

References

[1]	R. A. Adams, J. J. F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, 140 (2003).
[2]	G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Springer Monographs in Mathematics, Springer, New York, 2 (2011).
[3]	L. Grafakos, S. Oh, The Kato-Ponce inequality, Commun. Part. Diff. Equ., 39 (2014), 1128–1157. https://doi.org/10.1080/03605302.2013.822885
[4]	B. H. Haak, P. C. Kunstmann, On Kato's method for Navier-Stokes equations, J. Math. Fluid Mech., 11 (2009), 492–535. https://doi.org/10.1007/s00021-008-0270-5 doi: 10.1007/s00021-008-0270-5
[5]	M. Haase, The functional calculus for sectorial operators, Oper. Theory Adv. Appl., 169 (2006).
[6]	D. Jerison, C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161–219. https://doi.org/10.1006/jfan.1995.1067 doi: 10.1006/jfan.1995.1067
[7]	J. Lang, O. Méndez, Potential techniques and regularity of boundary value problems in exterior non-smooth domains: Regularity in exterior domains, Potential Anal., 24 (2006), 385–406. https://doi.org/10.1007/s11118-006-9008-2 doi: 10.1007/s11118-006-9008-2
[8]	D. Mitrea, Layer potentials and Hodge decompositions in two dimensional Lipschitz domains, Math. Ann., 322 (2002), 75–101. https://doi.org/10.1007/s002080100266 doi: 10.1007/s002080100266
[9]	T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115–140. https://doi.org/10.32917/hmj/1206133879 doi: 10.32917/hmj/1206133879
[10]	Y. Shibata, On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain, Commun. Pure Appl. Anal., 17 (2018), 1681–1721. https://doi.org/10.3934/cpaa.2018081 doi: 10.3934/cpaa.2018081
[11]	C. G. Simader, H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, Series on Advances in Mathematics for Applied Sciences, Mathematical Problems Relating to the Navier-Stokes Equations, River Edge, NJ, 11 (1992), 1–35. https://doi.org/10.1142/9789814503594_0001
[12]	P. Tolksdorf, K. Watanabe, The Navier-Stokes equations in exterior Lipschitz domains: $L^p$-theory, J. Differ. Equations, 269 (2020), 5765–5801. https://doi.org/10.1016/j.jde.2020.04.015 doi: 10.1016/j.jde.2020.04.015
[13]	K. Watanabe, Decay estimates of gradient of the Stokes semigroup in exterior Lipschitz domains, J. Differ. Equations, 346 (2023), 277–312. https://doi.org/10.1016/j.jde.2022.11.045 doi: 10.1016/j.jde.2022.11.045
[14]	M. Wilke, On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations in cylindrical domains, Interface. Free Bound., 24 (2022), 487–531. https://doi.org/10.4171/ifb/480 doi: 10.4171/ifb/480
[15]	I. Wood, Maximal $L^p$-regularity for the Laplacian on Lipschitz domains, Math. Z., 255 (2007), 855–875. https://doi.org/10.1007/s00209-006-0055-6 doi: 10.1007/s00209-006-0055-6

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)