July 2018, 38(7): 3357-3385. doi: 10.3934/dcds.2018144

Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received  June 2017 Revised  January 2018 Published  April 2018

Fund Project: The first author was supported by the NNSF of China (No. 11501021), the second author was supported by the NNSF of China (No. 11301166)

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in Hölder spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.

Citation: Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144
References:
[1]

C. J. Amick and R. E. L. Turner, A global theory of internal solitary waves in two-fluid systems, Trans. Amer. Math. Soc., 298 (1986), 431-484. doi: 10.1090/S0002-9947-1986-0860375-3.

[2]

C. J. Amick, Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 441-499.

[3]

D. E. Apushkinskaya and A. I. Nazarov, A survey of results on nonlinear Venttsel problems, Appl. Math., 45 (2000), 69-80. doi: 10.1023/A:1022288717033.

[4]

D. E. Apushkinskaya and A. I. Nazarov, Linear two-phase Venttsel problems, Ark. Mat., 39 (2001), 201-222. doi: 10.1007/BF02384554.

[5]

A. A. Arkhipova and O. Erlhamahmy, Regularity of solutions to a diffraction-type problem for nondiagonal linear elliptic systems in the Campanato space, J. Math. Sci. (New York), 112 (2002), 3944-3966. doi: 10.1023/A:1020093606080.

[6]

J. Thomas BealeT. Y. Hou and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., 46 (1993), 1269-1301. doi: 10.1002/cpa.3160460903.

[7]

J. Bognár, Indefinite Inner Product Spaces, Springer-Verlag, New York-Heidelberg, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78.

[8]

J. L. BonaD. K. Bose and R. E. L. Turner, Finite-amplitude steady waves in stratified fluids, J. Math. Pures Appl. (9), 62 (1983), 389-439(1984).

[9]

V. Bonnaillie-NoëlM. DambrineF. Hérau and G. Vial, On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal., 42 (2010), 931-945. doi: 10.1137/090756521.

[10]

M. Borsuk, The transmission problem for elliptic second order equations in a conical domain, Ann. Acad. Pedagog. Crac. Stud. Math., 7 (2008), 61-89.

[11]

M. Borsuk, The transmission problem for quasi-linear elliptic second order equations in a conical domain. Ⅰ, Ⅱ, Nonlinear Anal., 71 (2009), 5032-5083. doi: 10.1016/j.na.2009.03.090.

[12]

M. Borsuk, Transmission Problems for Elliptic Second-order Equations in Non-Smooth Domains, Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2010. doi: 10.1007/978-3-0346-0477-2.

[13]

O. BühlerJ. Shatah and S. Walsh, Steady water waves in the presence of wind, SIAM J. Math. Anal., 45 (2013), 2182-2227. doi: 10.1137/120880124.

[14]

O. BühlerJ. ShatahS. Walsh and C. Zeng, On the wind generation of water waves, Arch. Ration. Mech. Anal., 222 (2016), 827-878. doi: 10.1007/s00205-016-1012-0.

[15]

R. M. Chen and S. Walsh, Continuous dependence on the density for stratified steady water waves, Arch. Ration. Mech. Anal., 219 (2016), 741-792. doi: 10.1007/s00205-015-0906-6.

[16]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[17]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[18]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[19]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[20]

M. -L. Dubreil-Jacotin, Sur la Détermination Rigoureuse des Ondes Permanentes périodiques D'ampleur Finie, (French) 1934. 75 pp.

[21]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[22]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, volume 24 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989.

[23]

I. S. Iohvidov, M. G. Krein and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, volume 9 of Mathematical Research, Akademie-Verlag, Berlin, 1982.

[24]

K. Kirchgässner, Wave-solutions of reversible systems and applications, J. Differential Equations, 45 (1982), 113-127. doi: 10.1016/0022-0396(82)90058-4.

[25]

P. Korman, Existence of solutions for a class of semilinear noncoercive problems, Nonlinear Anal., 10 (1986), 1471-1476. doi: 10.1016/0362-546X(86)90116-1.

[26]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.

[27]

K. Lankers and G. Friesecke, Fast, large-amplitude solitary waves in the 2D Euler equations for stratified fluids, Nonlinear Anal., 29 (1997), 1061-1078. doi: 10.1016/S0362-546X(96)00089-2.

[28]

Y. Luo, On the quasilinear elliptic Venttsel$ \prime $ boundary value problem, Nonlinear Anal., 16 (1991), 761-769. doi: 10.1016/0362-546X(91)90081-B.

[29]

Y. Luo and N. S. Trudinger, Linear second order elliptic equations with Venttsel boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 193-207. doi: 10.1017/S0308210500029048.

[30]

Y. Luo and N. S. Trudinger, Quasilinear second order elliptic equations with Venttsel boundary conditions, Potential Anal., 3 (1994), 219-243. doi: 10.1007/BF01053434.

[31]

C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114. doi: 10.1016/j.jde.2014.01.036.

[32]

A.-V. Matioc and B.-V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886. doi: 10.1007/s00220-014-1918-z.

[33]

J. W. Miles, On the generation of surface waves by shear flows, J. Fluid Mech., 3 (1957), 185-204. doi: 10.1017/S0022112057000567.

[34]

A. I. Nazarov and A. A. Paletskikh, On the Hölder property of the solutions of the elliptic Venttsel problem, Dokl. Akad. Nauk, 465 (2015), 532-536.

[35]

D. V. Nilsson, Internal gravity-capillary solitary waves in finite depth, Math. Methods Appl. Sci., 40 (2017), 1053-1080. doi: 10.1002/mma.4036.

[36]

O. A. Oleǐnik, Boundary-value problems for linear equations of elliptic parabolic type with discontinuous coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 3-20.

[37]

M. Schechter, A generalization of the problem of transmission, Ann. Scuola Norm. Sup. Pisa (3), 14 (1960), 207-236.

[38]

Z. G. Šeftel', Estimates in $ L_{p} $ of solutions of elliptic equations with discontinuous coefficients and satisfying general boundary conditions and conjugacy conditions, Soviet Math. Dokl., 4 (1963), 321-324.

[39]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[40]

S. M. Sun, Existence of solitary internal waves in a two-layer fluid of infinite depth, In Proceedings of the Second World Congress of Nonlinear Analysts, Part 8 (Athens, 1996), 30 (1997), 5481-5490. doi: 10.1016/S0362-546X(97)00178-8.

[41]

S. M. Sun, Solitary internal waves in continuously stratified fluids of great depth, Phys. D, 166 (2002), 76-103. doi: 10.1016/S0167-2789(02)00424-4.

[42]

R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 513-573.

[43]

A. D. Ventcel', On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177. doi: 10.1137/1104014.

[44]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943. doi: 10.1137/050630465.

[45]

E. Wahlén, Steady periodic capillary waves with vorticity, Ark. Mat., 44 (2006), 367-387. doi: 10.1007/s11512-006-0024-7.

[46]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[47]

S. Walsh, Steady stratified periodic gravity waves with surface tension Ⅰ: Local bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3241-3285. doi: 10.3934/dcds.2014.34.3241.

[48]

S. Walsh, Steady stratified periodic gravity waves with surface tension Ⅱ: global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315. doi: 10.3934/dcds.2014.34.3287.

[49]

J. R. Wilton, On ripples, Phil. Mag., 1915, 29pp.

show all references

References:
[1]

C. J. Amick and R. E. L. Turner, A global theory of internal solitary waves in two-fluid systems, Trans. Amer. Math. Soc., 298 (1986), 431-484. doi: 10.1090/S0002-9947-1986-0860375-3.

[2]

C. J. Amick, Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 441-499.

[3]

D. E. Apushkinskaya and A. I. Nazarov, A survey of results on nonlinear Venttsel problems, Appl. Math., 45 (2000), 69-80. doi: 10.1023/A:1022288717033.

[4]

D. E. Apushkinskaya and A. I. Nazarov, Linear two-phase Venttsel problems, Ark. Mat., 39 (2001), 201-222. doi: 10.1007/BF02384554.

[5]

A. A. Arkhipova and O. Erlhamahmy, Regularity of solutions to a diffraction-type problem for nondiagonal linear elliptic systems in the Campanato space, J. Math. Sci. (New York), 112 (2002), 3944-3966. doi: 10.1023/A:1020093606080.

[6]

J. Thomas BealeT. Y. Hou and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., 46 (1993), 1269-1301. doi: 10.1002/cpa.3160460903.

[7]

J. Bognár, Indefinite Inner Product Spaces, Springer-Verlag, New York-Heidelberg, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78.

[8]

J. L. BonaD. K. Bose and R. E. L. Turner, Finite-amplitude steady waves in stratified fluids, J. Math. Pures Appl. (9), 62 (1983), 389-439(1984).

[9]

V. Bonnaillie-NoëlM. DambrineF. Hérau and G. Vial, On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal., 42 (2010), 931-945. doi: 10.1137/090756521.

[10]

M. Borsuk, The transmission problem for elliptic second order equations in a conical domain, Ann. Acad. Pedagog. Crac. Stud. Math., 7 (2008), 61-89.

[11]

M. Borsuk, The transmission problem for quasi-linear elliptic second order equations in a conical domain. Ⅰ, Ⅱ, Nonlinear Anal., 71 (2009), 5032-5083. doi: 10.1016/j.na.2009.03.090.

[12]

M. Borsuk, Transmission Problems for Elliptic Second-order Equations in Non-Smooth Domains, Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2010. doi: 10.1007/978-3-0346-0477-2.

[13]

O. BühlerJ. Shatah and S. Walsh, Steady water waves in the presence of wind, SIAM J. Math. Anal., 45 (2013), 2182-2227. doi: 10.1137/120880124.

[14]

O. BühlerJ. ShatahS. Walsh and C. Zeng, On the wind generation of water waves, Arch. Ration. Mech. Anal., 222 (2016), 827-878. doi: 10.1007/s00205-016-1012-0.

[15]

R. M. Chen and S. Walsh, Continuous dependence on the density for stratified steady water waves, Arch. Ration. Mech. Anal., 219 (2016), 741-792. doi: 10.1007/s00205-015-0906-6.

[16]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[17]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[18]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[19]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[20]

M. -L. Dubreil-Jacotin, Sur la Détermination Rigoureuse des Ondes Permanentes périodiques D'ampleur Finie, (French) 1934. 75 pp.

[21]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[22]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, volume 24 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989.

[23]

I. S. Iohvidov, M. G. Krein and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, volume 9 of Mathematical Research, Akademie-Verlag, Berlin, 1982.

[24]

K. Kirchgässner, Wave-solutions of reversible systems and applications, J. Differential Equations, 45 (1982), 113-127. doi: 10.1016/0022-0396(82)90058-4.

[25]

P. Korman, Existence of solutions for a class of semilinear noncoercive problems, Nonlinear Anal., 10 (1986), 1471-1476. doi: 10.1016/0362-546X(86)90116-1.

[26]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.

[27]

K. Lankers and G. Friesecke, Fast, large-amplitude solitary waves in the 2D Euler equations for stratified fluids, Nonlinear Anal., 29 (1997), 1061-1078. doi: 10.1016/S0362-546X(96)00089-2.

[28]

Y. Luo, On the quasilinear elliptic Venttsel$ \prime $ boundary value problem, Nonlinear Anal., 16 (1991), 761-769. doi: 10.1016/0362-546X(91)90081-B.

[29]

Y. Luo and N. S. Trudinger, Linear second order elliptic equations with Venttsel boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 193-207. doi: 10.1017/S0308210500029048.

[30]

Y. Luo and N. S. Trudinger, Quasilinear second order elliptic equations with Venttsel boundary conditions, Potential Anal., 3 (1994), 219-243. doi: 10.1007/BF01053434.

[31]

C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114. doi: 10.1016/j.jde.2014.01.036.

[32]

A.-V. Matioc and B.-V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886. doi: 10.1007/s00220-014-1918-z.

[33]

J. W. Miles, On the generation of surface waves by shear flows, J. Fluid Mech., 3 (1957), 185-204. doi: 10.1017/S0022112057000567.

[34]

A. I. Nazarov and A. A. Paletskikh, On the Hölder property of the solutions of the elliptic Venttsel problem, Dokl. Akad. Nauk, 465 (2015), 532-536.

[35]

D. V. Nilsson, Internal gravity-capillary solitary waves in finite depth, Math. Methods Appl. Sci., 40 (2017), 1053-1080. doi: 10.1002/mma.4036.

[36]

O. A. Oleǐnik, Boundary-value problems for linear equations of elliptic parabolic type with discontinuous coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 3-20.

[37]

M. Schechter, A generalization of the problem of transmission, Ann. Scuola Norm. Sup. Pisa (3), 14 (1960), 207-236.

[38]

Z. G. Šeftel', Estimates in $ L_{p} $ of solutions of elliptic equations with discontinuous coefficients and satisfying general boundary conditions and conjugacy conditions, Soviet Math. Dokl., 4 (1963), 321-324.

[39]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[40]

S. M. Sun, Existence of solitary internal waves in a two-layer fluid of infinite depth, In Proceedings of the Second World Congress of Nonlinear Analysts, Part 8 (Athens, 1996), 30 (1997), 5481-5490. doi: 10.1016/S0362-546X(97)00178-8.

[41]

S. M. Sun, Solitary internal waves in continuously stratified fluids of great depth, Phys. D, 166 (2002), 76-103. doi: 10.1016/S0167-2789(02)00424-4.

[42]

R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 513-573.

[43]

A. D. Ventcel', On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177. doi: 10.1137/1104014.

[44]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943. doi: 10.1137/050630465.

[45]

E. Wahlén, Steady periodic capillary waves with vorticity, Ark. Mat., 44 (2006), 367-387. doi: 10.1007/s11512-006-0024-7.

[46]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[47]

S. Walsh, Steady stratified periodic gravity waves with surface tension Ⅰ: Local bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3241-3285. doi: 10.3934/dcds.2014.34.3241.

[48]

S. Walsh, Steady stratified periodic gravity waves with surface tension Ⅱ: global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315. doi: 10.3934/dcds.2014.34.3287.

[49]

J. R. Wilton, On ripples, Phil. Mag., 1915, 29pp.

Figure 1.  The air-water system
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