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Traveling wave solutions to the multilayer free boundary incompressible Navier-Stokes equations. (English) Zbl 1479.35634

Summary: For a natural number \(m \geq 2\), we study \(m\) layers of finite depth, horizontally infinite, viscous, and incompressible fluid bounded below by a flat rigid bottom. Adjacent layers meet at free interface regions, and the top layer is bounded above by a free boundary as well. A uniform gravitational field, normal to the rigid bottom, acts on the fluid. We assume that the fluid mass densities are strictly decreasing from bottom to top and consider the cases with and without surface tension acting on the free surfaces. In addition to these gravity-capillary effects, we allow a force to act on the bulk and external stress tensors to act on the free interface regions. Both of these additional forces are posited to be in traveling wave form: time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. Without surface tension in the case of two dimensional fluids and with all positive surface tensions in the higher dimensional cases, we prove that for each sufficiently small force and stress tuple there exists a traveling wave solution. The existence of traveling wave solutions to the one layer configuration \((m=1)\) was recently established and, to the best of our knowledge, this paper is the first construction of traveling wave solutions to the incompressible Navier-Stokes equations in the \(m\)-layer arrangement.

MSC:

35Q30 Navier-Stokes equations
35R35 Free boundary problems for PDEs
35C07 Traveling wave solutions
35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
76D33 Waves for incompressible viscous fluids
76D45 Capillarity (surface tension) for incompressible viscous fluids
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[1] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Appl. Math. Sci. 75, 2nd ed., Springer, New York, 1988, https://doi.org/10.1007/978-1-4612-1029-0. · Zbl 0875.58002
[2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), pp. 35-92, https://doi.org/10.1002/cpa.3160170104. · Zbl 0123.28706
[3] H. Bae and K. Cho, Free surface problem of stationary non-Newtonian fluids, Nonlinear Anal., 41 (2000), pp. 243-258, https://doi.org/10.1016/S0362-546X(98)00276-4. · Zbl 0967.76004
[4] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), pp. 359-392 · Zbl 0464.76028
[5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. · Zbl 1220.46002
[6] D. Chae and P. Dubovskiǐ, Traveling wave-like solutions of the Navier-Stokes and the related equations, J. Math. Anal. Appl., 204 (1996), pp. 930-939. · Zbl 0874.35084
[7] Y. Cho, J. D. Diorio, T. R. Akylas, and J. H. Duncan, Resonantly forced gravity-capillary lumps on deep water. Part 2. Theoretical model, J. Math. Fluid Mech., 672 (2011), pp. 288-306. · Zbl 1225.76059
[8] J. D. Diorio, Y. Cho, J. H. Duncan, and T. R. Akylas, Resonantly forced gravity-capillary lumps on deep water. Part 1. Experiments, J. Math. Fluid Mech., 672 (2011), pp. 268-287. · Zbl 1225.76060
[9] G. B. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. · Zbl 0841.35001
[10] R. S. Gellrich, Free boundary value problems for the stationary Navier-Stokes equations in domains with noncompact boundaries, Z. Anal. Anwend., 12 (1993), pp. 425-455, https://doi.org/10.4171/ZAA/554. · Zbl 0778.76019
[11] L. Grafakos, Classical Fourier Analysis, Grad. Texts Math. 249, 3rd ed., Springer, New York, 2014, https://doi.org/10.1007/978-1-4939-1194-3. · Zbl 1304.42001
[12] M. D. Groves, Steady water waves, J. Nonlinear Math. Phys., 11 (2004), pp. 435-460, https://doi.org/10.2991/jnmp.2004.11.4.2. · Zbl 1064.35144
[13] M. Jean, Free surface of the steady flow of a Newtonian fluid in a finite channel, Arch. Ration. Mech. Anal., 74 (1980), pp. 197-217. · Zbl 0451.76038
[14] Y. Kagei and T. Nishida, Traveling waves bifurcating from plane Poiseuille flow of the compressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 231 (2019), pp. 1-44, https://doi.org/10.1007/s00205-018-1269-6. · Zbl 1419.76539
[15] G. Leoni and I. Tice, Traveling wave solutions to the free boundary incompressible Navier-Stokes equations, Comm. Pure Appl. Math., to appear. · Zbl 07749389
[16] N. Masnadi and J. H. Duncan, The generation of gravity-capillary solitary waves by a pressure source moving at a trans-critical speed, J. Fluid Mech., 810 (2017), pp. 448-474, https://doi.org/10.1017/jfm.2016.658.
[17] S. A. Nazarov and K. Pileckas, On the solvability of the Stokes and Navier-Stokes problems in the domains that are layer-like at infinity, J. Math. Fluid Mech., 1 (1999), pp. 78-116, https://doi.org/10.1007/s000210050005. · Zbl 0941.35062
[18] S. A. Nazarov and K. Pileckas, The asymptotic properties of the solution to the Stokes problem in domains that are layer-like at infinity, J. Math. Fluid Mech., 1 (1999), pp. 131-167, https://doi.org/10.1007/s000210050007. · Zbl 0940.35155
[19] B. Park and Y. Cho, Experimental observation of gravity-capillary solitary waves generated by a moving air suction, J. Math. Fluid Mech., 808 (2016), pp. 168-188. · Zbl 1383.76011
[20] B. Park and Y. Cho, Two-dimensional gravity-capillary solitary waves on deep water: Generation and transverse instability, J. Math. Fluid Mech., 834 (2018), pp. 92-124.
[21] K. Pileckas, Gliding of a flat plate of infinite span over the surface of a heavy viscous incompressible fluid of finite depth, Differ. Uravn. Primen., (1983), pp. 60-74. · Zbl 0564.76052
[22] K. Pileckas, A remark on the paper: “Gliding of a flat plate of infinite span over the surface of a heavy viscous incompressible fluid of finite depth” , Differ. Uravn. Primen., (1984), pp. 55-60, 139. · Zbl 0588.76046
[23] K. Pileckas, On the asymptotic behavior of solutions of a stationary system of Navier-Stokes equations in a domain of layer type, Mat. Sb., 193 (2002), pp. 69-104, https://doi.org/10.1070/SM2002v193n12ABEH000700. · Zbl 1069.35052
[24] K. Pileckas and L. Zaleskis, On a steady three-dimensional noncompact free boundary value problem for the Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 306 (2003), pp. 134-164. · Zbl 1148.35357
[25] W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), pp. 671-694, https://doi.org/10.1090/S0273-0979-2010-01302-1. · Zbl 1426.76078
[26] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), pp. 1-48. · Zbl 0897.35067
[27] J. V. Wehausen and E. V. Laitone, Surface waves, in Handbuch der Physik, Vol. 9, Part 3, Springer, Berlin, 1960, pp. 446-778. · Zbl 1339.76009
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