## Analysis & PDE

• Anal. PDE
• Volume 6, Number 2 (2013), 287-369.

### Local well-posedness of the viscous surface wave problem without surface tension

#### Abstract

We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The domain is allowed to have a horizontal cross-section that is either periodic or infinite in extent. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. This paper is the first in a series of three on the global well-posedness and decay of the viscous surface wave problem without surface tension. Here we develop a local well-posedness theory for the equations in the framework of the nonlinear energy method, which is based on the natural energy structure of the problem. Our proof involves several novel techniques, including: energy estimates in a “geometric” reformulation of the equations, a well-posedness theory of the linearized Navier–Stokes equations in moving domains, and a time-dependent functional framework, which couples to a Galerkin method with a time-dependent basis.

#### Article information

Source
Anal. PDE, Volume 6, Number 2 (2013), 287-369.

Dates
Revised: 7 March 2012
Accepted: 23 May 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731312

Digital Object Identifier
doi:10.2140/apde.2013.6.287

Mathematical Reviews number (MathSciNet)
MR3071393

Zentralblatt MATH identifier
1273.35209

#### Citation

Guo, Yan; Tice, Ian. Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6 (2013), no. 2, 287--369. doi:10.2140/apde.2013.6.287. https://projecteuclid.org/euclid.apde/1513731312

#### References

• H. Abels, “The initial-value problem for the Navier–Stokes equations with a free surface in $L\sp q$-Sobolev spaces”, Adv. Differential Equations 10:1 (2005), 45–64.
• R. A. Adams, Sobolev spaces, Pure and Applied Mathematics 65, Academic Press, New York, 1975.
• S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I”, Comm. Pure Appl. Math. 12 (1959), 623–727.
• 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), 35–92.
• H. Bae, “Solvability of the free boundary value problem of the Navier–Stokes equations”, Discrete Contin. Dyn. Syst. 29:3 (2011), 769–801.
• J. T. Beale, “The initial value problem for the Navier–Stokes equations with a free surface”, Comm. Pure Appl. Math. 34:3 (1981), 359–392.
• J. T. Beale, “Large-time regularity of viscous surface waves”, Arch. Rational Mech. Anal. 84:4 (1984), 307–352.
• J. T. Beale and T. Nishida, “Large-time behavior of viscous surface waves”, pp. 1–14 in Recent topics in nonlinear PDE, II (Sendai, 1984), edited by K. Masuda and M. Mimura, North-Holland Mathematics Studies 128, North-Holland Publishing, Amsterdam, 1985.
• J. P. Bourguignon and H. Brezis, “Remarks on the Euler equation”, J. Functional Analysis 15 (1974), 341–363.
• C. H. A. Cheng and S. Shkoller, “The interaction of the 3D Navier–Stokes equations with a moving nonlinear Koiter elastic shell”, SIAM J. Math. Anal. 42:3 (2010), 1094–1155.
• D. Christodoulou and H. Lindblad, “On the motion of the free surface of a liquid”, Comm. Pure Appl. Math. 53:12 (2000), 1536–1602.
• D. Coutand and S. Shkoller, “Unique solvability of the free-boundary Navier–Stokes equations with surface tension”, preprint, 2003.
• D. Coutand and S. Shkoller, “Well-posedness of the free-surface incompressible Euler equations with or without surface tension”, J. Amer. Math. Soc. 20:3 (2007), 829–930.
• R. Danchin, “Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients”, Rev. Mat. Iberoamericana 21:3 (2005), 863–888.
• R. Danchin, “Fourier analysis methods for PDEs”, preprint, UMR 8050, 2005, http://perso-math.univ-mlv.fr/users/danchin.raphael/recherche.html.
• L. C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010.
• P. Germain, N. Masmoudi, and J. Shatah, “Global solutions for the gravity water waves equation in dimension 3”, C. R. Math. Acad. Sci. Paris 347:15-16 (2009), 897–902.
• Y. Guo and I. Tice, “Decay of viscous surface waves without surface tension”, preprint, 2010. http://msp.org/idx/arx/1011.5179arXiv 1011.5179
• Y. Guo and I. Tice, “Almost exponential decay of periodic viscous surface waves without surface tension”, Arch. Ration. Mech. Anal. 207:2 (2013), 459–531.
• Y. Guo and I. Tice, “Decay of viscous surface waves without surface tension in horizontally infinite domains”, 6 (2013). To appear.
• Y. Hataya, “Decaying solution of a Navier–Stokes flow without surface tension”, J. Math. Kyoto Univ. 49:4 (2009), 691–717.
• D. Lannes, “Well-posedness of the water-waves equations”, J. Amer. Math. Soc. 18:3 (2005), 605–654.
• H. Lindblad, “Well-posedness for the motion of an incompressible liquid with free surface boundary”, Ann. of Math. $(2)$ 162:1 (2005), 109–194.
• T. Nishida, Y. Teramoto, and H. Yoshihara, “Global in time behavior of viscous surface waves: Horizontally periodic motion”, J. Math. Kyoto Univ. 44:2 (2004), 271–323.
• J. Shatah and C. Zeng, “Geometry and a priori estimates for free boundary problems of the Euler equation”, Comm. Pure Appl. Math. 61:5 (2008), 698–744.
• V. A. Solonnikov, “Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface”, Izv. Akad. Nauk SSSR Ser. Mat. 41:6 (1977), 1388–1424. In Russian; translated in Math. USSR-Izv 11 (1977):6 (1978), 1323–1358.
• V. A. Solonnikov and V. E. Skadilov, “On a boundary value problem for a stationary system of Navier–Stokes equations”, Proc. Steklov Inst. Math. 125 (1973), 186–199.
• D. L. G. Sylvester, “Large time existence of small viscous surface waves without surface tension”, Comm. Partial Differential Equations 15:6 (1990), 823–903.
• A. Tani and N. Tanaka, “Large-time existence of surface waves in incompressible viscous fluids with or without surface tension”, Arch. Rational Mech. Anal. 130:4 (1995), 303–314.
• S. Wu, “Well-posedness in Sobolev spaces of the full water wave problem in $2$-D”, Invent. Math. 130:1 (1997), 39–72.
• S. Wu, “Well-posedness in Sobolev spaces of the full water wave problem in 3-D”, J. Amer. Math. Soc. 12:2 (1999), 445–495.
• S. Wu, “Almost global wellposedness of the 2-D full water wave problem”, Invent. Math. 177:1 (2009), 45–135.
• S. Wu, “Global wellposedness of the 3-D full water wave problem”, Invent. Math. 184:1 (2011), 125–220.
• P. Zhang and Z. Zhang, “On the free boundary problem of three-dimensional incompressible Euler equations”, Comm. Pure Appl. Math. 61:7 (2008), 877–940.