A comparison theorem for super- and subsolutions of $\mathbf {\nabla ^2 u + f (u) = 0}$ and its application to water waves with vorticity
HTML articles powered by AMS MathViewer
- by V. Kozlov and N. Kuznetsov
- St. Petersburg Math. J. 30 (2019), 471-483
- DOI: https://doi.org/10.1090/spmj/1554
- Published electronically: April 12, 2019
- PDF | Request permission
Abstract:
A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form $f(u)$ with $f$ belonging to the class $L^p_{\operatorname {loc}}$ and the solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.References
- Jürgen Appell and Petr P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, Cambridge, 1990. MR 1066204, DOI 10.1017/CBO9780511897450
- Constantin Carathéodory, Vorlesungen über reelle Funktionen, Chelsea Publishing Co., New York, 1968 (German). Third (corrected) edition. MR 0225940
- Adrian Constantin and Walter Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481–527. MR 2027299, DOI 10.1002/cpa.3046
- Adrian Constantin and Walter Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 133–175. MR 2835865, DOI 10.1007/s00205-011-0412-4
- Adrian Constantin, Walter Strauss, and Eugen Vărvărucă, Global bifurcation of steady gravity water waves with critical layers, Acta Math. 217 (2016), no. 2, 195–262. MR 3689941, DOI 10.1007/s11511-017-0144-x
- M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie, NUMDAM, [place of publication not identified], 1934 (French). MR 3533020
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math. 10 (1957), 503–510. MR 91407, DOI 10.1002/cpa.3160100402
- V. Kozlov and N. Kuznetsov, Steady free-surface vortical flows parallel to the horizontal bottom, Quart. J. Mech. Appl. Math. 64 (2011), no. 3, 371–399. MR 2847132, DOI 10.1093/qjmam/hbr010
- Vladimir Kozlov and Nikolay Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 971–1018. MR 3269640, DOI 10.1007/s00205-014-0787-0
- V. Kozlov and N. Kuznetsov, Bounds for solutions to the problem of steady water waves with vorticity, Quart. J. Mech. Appl. Math. 70 (2017), no. 4, 497–518. MR 3737353, DOI 10.1093/qjmam/hbx019
- V. Kozlov, N. Kuznetsov, and E. Lokharu, On bounds and non-existence in the problem of steady waves with vorticity, J. Fluid Mech. 765 (2015), R1, 13. MR 3300703, DOI 10.1017/jfm.2014.747
- V. Kozlov, N. Kuznetsov, and E. Lokharu, On the Benjamin-Lighthill conjecture for water waves with vorticity, J. Fluid Mech. 825 (2017), 961–1001. MR 3679011, DOI 10.1017/jfm.2017.361
- Kazuhiro Kurata, Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order, Indiana Univ. Math. J. 43 (1994), no. 2, 411–440. MR 1291523, DOI 10.1512/iumj.1994.43.43017
- M. A. Lavrent′ev and B. V. Shabat, Problemy gidrodinamiki i ikh matematicheskie modeli, 2nd ed., Izdat. “Nauka”, Moscow, 1977 (Russian). MR 0459198
- Calin Iulian Martin and Bogdan-Vasile Matioc, Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results, J. Nonlinear Sci. 24 (2014), no. 4, 633–659. MR 3228471, DOI 10.1007/s00332-014-9201-1
- Robert Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641–1647. MR 98239
- Walter A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 671–694. MR 2721042, DOI 10.1090/S0273-0979-2010-01302-1
- José C. Sabina de Lis, Hopf maximum principle revisited, Electron. J. Differential Equations (2015), No. 115, 9. MR 3358487
- Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406
- J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth, IMA J. Appl. Math. 56 (1996), 207–217.
Bibliographic Information
- V. Kozlov
- Affiliation: Department of Mathematics, Linköping University, S–581 83 Linköping, Sweden
- Email: vladimir.kozlov@liu.se
- N. Kuznetsov
- Affiliation: Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol′shoy pr. 61, V.O., 199178 St. Petersburg, Russia
- MR Author ID: 242194
- Email: nikolay.g.kuznetsov@gmail.com
- Received by editor(s): October 10, 2017
- Published electronically: April 12, 2019
- Additional Notes: The first author was supported by the Swedish Research Council (VR) through the grant [EO418401].
The second author acknowledges the support from the G. S. Magnuson’s Foundation of the Royal Swedish Academy of Sciences and Linköping University. - © Copyright 2019 American Mathematical Society
- Journal: St. Petersburg Math. J. 30 (2019), 471-483
- MSC (2010): Primary 35P99; Secondary 76B15, 35Q35
- DOI: https://doi.org/10.1090/spmj/1554
- MathSciNet review: 3812002
Dedicated: In memoriam of M. Z. Solomyak