Stream functions for divergence-free vector fields
Author:
James P. Kelliher
Journal:
Quart. Appl. Math. 79 (2021), 163-174
MSC (2010):
Primary 35F15, 35Q35; Secondary 26B12
DOI:
https://doi.org/10.1090/qam/1575
Published electronically:
June 18, 2020
MathSciNet review:
4188627
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Abstract: In 1990, von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector field $u$ in a 3D bounded domain that is tangential to the boundary can be written as the curl of a vector field vanishing on the boundary of the domain. We extend this result to higher dimension and to Lipschitz boundaries in a form suitable for integration in flat space, showing that $u$ can be written as the divergence of an antisymmetric matrix field. We also demonstrate how obtaining a kernel for such a matrix field is dual to obtaining a Biot-Savart kernel for the domain.
References
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- G. Schwarz, The existence of solutions of a general boundary value problem for the divergence, Math. Methods Appl. Sci. 17 (1994), no. 2, 95–105. MR 1258258, DOI https://doi.org/10.1002/mma.1670170203
- Günter Schwarz, Hodge decomposition—a method for solving boundary value problems, Lecture Notes in Mathematics, vol. 1607, Springer-Verlag, Berlin, 1995. MR 1367287
- Roger Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis; Reprint of the 1984 edition. MR 1846644
- Wolf von Wahl, On necessary and sufficient conditions for the solvability of the equations ${\rm rot}\,u=\gamma $ and ${\rm div}\,u=\epsilon $ with $u$ vanishing on the boundary, The Navier-Stokes equations (Oberwolfach, 1988) Lecture Notes in Math., vol. 1431, Springer, Berlin, 1990, pp. 152–157. MR 1072185, DOI https://doi.org/10.1007/BFb0086065
References
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI https://doi.org/10.1002/%28SICI%291099-1476%28199806%2921%3A9%24%5Clangle%24823%3A%3AAID-MMA976%24%5Crangle%243.0.CO%3B2-B
- Chérif Amrouche, Philippe G. Ciarlet, and Cristinel Mardare, On a lemma of Jacques-Louis Lions and its relation to other fundamental results, J. Math. Pures Appl. (9) 104 (2015), no. 2, 207–226 (English, with English and French summaries). MR 3365827, DOI https://doi.org/10.1016/j.matpur.2014.11.007
- R. Benedetti, R. Frigerio, and R. Ghiloni, The topology of Helmholtz domains, Expo. Math. 30 (2012), no. 4, 319–375. MR 2997828, DOI https://doi.org/10.1016/j.exmath.2012.09.001
- Luigi C. Berselli and Placido Longo, Classical solutions for the system ${\text {curl} v = g}$, with vanishing Dirichlet boundary conditions, Discrete Contin. Dyn. Syst. Ser. S 12 (2019), no. 2, 215–229. MR 3842318, DOI https://doi.org/10.3934/dcdss.2019015
- M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037–1040 (Russian). MR 553920
- M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators $\mathrm {div}$ and $\mathrm {grad}$, Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. 1, vol. 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5–40, 149 (Russian). MR 631691
- Wolfgang Borchers and Hermann Sohr, On the equations $\mathrm {rot} \mathbf {v}=\mathbf {g}$ and $\mathrm {div} \mathbf {u}=f$ with zero boundary conditions, Hokkaido Math. J. 19 (1990), no. 1, 67–87. MR 1039466, DOI https://doi.org/10.14492/hokmj/1381517172
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- Jason Cantarella, Dennis DeTurck, and Herman Gluck, Vector calculus and the topology of domains in 3-space, Amer. Math. Monthly 109 (2002), no. 5, 409–442. MR 1901496, DOI https://doi.org/10.2307/2695643
- Alberto Enciso, M. Ángeles García-Ferrero, and Daniel Peralta-Salas, The Biot-Savart operator of a bounded domain, J. Math. Pures Appl. (9) 119 (2018), 85–113 (English, with English and French summaries). MR 3862144, DOI https://doi.org/10.1016/j.matpur.2017.11.004
- G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR 2808162
- Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations: Theory and algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383
- Mihaela Ignatova, Gautam Iyer, James P. Kelliher, Robert L. Pego, and Arghir D. Zarnescu, Global existence for two extended Navier-Stokes systems, Commun. Math. Sci. 13 (2015), no. 1, 249–267. MR 3238147, DOI https://doi.org/10.4310/CMS.2015.v13.n1.a12
- James P. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary, Commun. Math. Sci. 6 (2008), no. 4, 869–880. MR 2511697
- Dorina Mitrea, Marius Mitrea, and Sylvie Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal. 7 (2008), no. 6, 1295–1333. MR 2425010, DOI https://doi.org/10.3934/cpaa.2008.7.1295
- G. Schwarz, The existence of solutions of a general boundary value problem for the divergence, Math. Methods Appl. Sci. 17 (1994), no. 2, 95–105. MR 1258258, DOI https://doi.org/10.1002/mma.1670170203
- Günter Schwarz, Hodge decomposition—a method for solving boundary value problems, Lecture Notes in Mathematics, vol. 1607, Springer-Verlag, Berlin, 1995. MR 1367287
- Roger Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis; Reprint of the 1984 edition. MR 1846644
- Wolf von Wahl, On necessary and sufficient conditions for the solvability of the equations $\mathrm {rot} u=\gamma$ and $\mathrm {div} u=\varepsilon$ with $u$ vanishing on the boundary, The Navier-Stokes equations (Oberwolfach, 1988) Lecture Notes in Math., vol. 1431, Springer, Berlin, 1990, pp. 152–157. MR 1072185, DOI https://doi.org/10.1007/BFb0086065
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Additional Information
James P. Kelliher
Affiliation:
Department of Mathematics, University of California, Riverside, 900 University Avenue, Riverside, California 92521
MR Author ID:
744311
Email:
kelliher@math.ucr.edu
Received by editor(s):
May 22, 2020
Published electronically:
June 18, 2020
Article copyright:
© Copyright 2020
Brown University
