February  2017, 37(2): 1039-1059. doi: 10.3934/dcds.2017043

Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid

Dipartimento di Matematica "Giuseppe Peano", Universitá degli Studi di Torino, Via Carlo Alberto 10,10123 Torino, Italy

Received  February 2015 Revised  July 2016 Published  November 2016

Using a singular perturbation based approach, we make rigorous the formal boundary layer asymptotic analysis of Turcotte, Spence and Bau from the early eighties for the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work. A key point in our proof is to establish the non-degeneracy of a special solution of the Painlevé-Ⅰ transcendent. To this end, we relate this problem to recent studies for the ground states of the focusing nonlinear Schrödinger equation in an annulus. We also relate our result to a particular case of the well known Lazer-McKenna conjecture from nonlinear analysis.

Citation: Christos Sourdis. Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1039-1059. doi: 10.3934/dcds.2017043
References:
[1] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge university press, 2007. doi: 10.1017/cbo9780511618260.
[2]

H. Berestycki and J. Wei, On least energy solutions to a semilinear elliptic equation in a strip, Discrete Contin. Dyn. Syst., 28 (2010), 1083-1099. doi: 10.3934/dcds.2010.28.1083.

[3]

J. Byeon and Y. Oshita, Uniqueness of standing waves for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 975-987. doi: 10.1017/S0308210507000236.

[4]

J. ByeonO. Kwon and Y. Oshita, Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 14 (2015), 825-842. doi: 10.3934/cpaa.2015.14.825.

[5]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351. doi: 10.1016/j.jde.2004.07.017.

[6]

E. N. Dancer and S. Yan, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Methods Appl. Anal., 15 (2008), 97-119. doi: 10.4310/MAA.2008.v15.n1.a9.

[7]

M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898. doi: 10.1512/iumj.1999.48.1596.

[8]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.

[9]

P. FelmerS. Martínez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-Δ u +u = u^p$ in an annulus, J. Differential Equations, 245 (2008), 1198-1209. doi: 10.1016/j.jde.2008.06.006.

[10]

A. S. Fokas, A. R. Its, A. A. Kapaev and V. Y. Novokshenov, Painlevé Transcendents: The Riemann-Hilbert Approach, American Mathematical Society, Providence, RI, 2006.

[11]

C. Gallo and D. Pelinovsky, On the Thomas-Fermi ground state in a harmonic potential, Asymptot. Anal., 73 (2011), 53-96. doi: 10.3233/ASY-2011-1034.

[12]

S. P. Hastings and W. C. Troy, On some conjectures of Turcotte, Spence, Bau, and Holmes, SIAM J. Math. Anal., 20 (1989), 634-642. doi: 10.1137/0520045.

[13]

S. P. Hastings and J. B. McLeod, Classical Methods in Ordinary Differential Equations. With Applications to Boundary Value Problems, American Mathematical Society, Providence, RI, 2012.

[14] P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.
[15]

H. Hofer, A note on the topological degree at a critical point of mountain pass type, Proc. Amer. Math. Soc., 90 (1984), 309-315. doi: 10.1090/S0002-9939-1984-0727256-0.

[16]

P. Holmes, On a second-order boundary value problem arising in combustion theory, Quart. Appl. Math., 40 (1982/83), 525-538.

[17]

P. Holmes and D. Spence, On a Painlevé-type boundary-value problem, Quart. J. Mech. Appl. Math., 37 (1984), 525-538. doi: 10.1093/qjmam/37.4.525.

[18]

Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $\mathbb{R}^N$ and Séré's non-degeneracy condition, Comm. Partial Differential Equations, 24 (1999), 563-598. doi: 10.1080/03605309908821434.

[19]

G. Karali and C. Sourdis, Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 131-170. doi: 10.1016/j.anihpc.2011.09.005.

[20]

G. Karali and C. Sourdis, Resonance phenomena in a singular perturbation problem in the case of exchange of stabilities, Comm. Partial Differential Equations, 37 (2012), 1620-1667. doi: 10.1080/03605302.2012.681333.

[21]

G. Karali and C. Sourdis, The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit, Arch. Ration. Mech. Anal., 217 (2015), 439-523. doi: 10.1007/s00205-015-0844-3.

[22]

N. KawanoE. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to $Δ u +K(|x|)u^p=0$ in $\mathbb{R}^N$, Funkcial. Ekvac., 36 (1993), 557-579. doi: 10.1619/fesi.52.343.

[23]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1568. doi: 10.1002/cpa.10049.

[24]

P. D. Miller, Applied Asymptotic Analysis, American Mathematical Society, Providence, RI, 2006.

[25]

S. Nakamura, A Remark on eigenvalue splittings for one-dimensional double-well Hamiltonians, Lett. Math. Phys., 11 (1986), 337-340. doi: 10.1007/BF00574159.

[26]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[27]

L. Recke and O. E. Omel'chenko, Boundary layer solutions to problems with infinite-dimensional singular and regular perturbations, J. Differential Equations, 245 (2008), 3806-3822. doi: 10.1016/j.jde.2008.01.017.

[28]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, Vol. 309. Springer-Verlag, Berlin-New York, 1973.

[29]

S. Schecter and C. Sourdis, Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations, J. Dynam. Differential Equations, 22 (2010), 629-655. doi: 10.1007/s10884-010-9171-4.

[30]

C. Sourdis and P. C. Fife, Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces, Adv. Differential Equations, 12 (2007), 623-668.

[31]

S. K. TinN. Kopell and C. K. R. T. Jones, Invariant manifolds and singularly perturbed boundary value problems, SIAM J. Numer. Anal., 31 (1994), 1558-1576. doi: 10.1137/0731081.

[32]

D. L. TurcotteD. A. Spence and H. H. Bau, Multiple solutions for natural convective flows in an internally heated, vertical channel with viscous dissipation and pressure work, Int. J. Heat Mass Transfer, 25 (1982), 699-706. doi: 10.1016/0017-9310(82)90175-2.

[33] W. Walter, Ordinary Differential Equations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.

show all references

References:
[1] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge university press, 2007. doi: 10.1017/cbo9780511618260.
[2]

H. Berestycki and J. Wei, On least energy solutions to a semilinear elliptic equation in a strip, Discrete Contin. Dyn. Syst., 28 (2010), 1083-1099. doi: 10.3934/dcds.2010.28.1083.

[3]

J. Byeon and Y. Oshita, Uniqueness of standing waves for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 975-987. doi: 10.1017/S0308210507000236.

[4]

J. ByeonO. Kwon and Y. Oshita, Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 14 (2015), 825-842. doi: 10.3934/cpaa.2015.14.825.

[5]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351. doi: 10.1016/j.jde.2004.07.017.

[6]

E. N. Dancer and S. Yan, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Methods Appl. Anal., 15 (2008), 97-119. doi: 10.4310/MAA.2008.v15.n1.a9.

[7]

M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898. doi: 10.1512/iumj.1999.48.1596.

[8]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.

[9]

P. FelmerS. Martínez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-Δ u +u = u^p$ in an annulus, J. Differential Equations, 245 (2008), 1198-1209. doi: 10.1016/j.jde.2008.06.006.

[10]

A. S. Fokas, A. R. Its, A. A. Kapaev and V. Y. Novokshenov, Painlevé Transcendents: The Riemann-Hilbert Approach, American Mathematical Society, Providence, RI, 2006.

[11]

C. Gallo and D. Pelinovsky, On the Thomas-Fermi ground state in a harmonic potential, Asymptot. Anal., 73 (2011), 53-96. doi: 10.3233/ASY-2011-1034.

[12]

S. P. Hastings and W. C. Troy, On some conjectures of Turcotte, Spence, Bau, and Holmes, SIAM J. Math. Anal., 20 (1989), 634-642. doi: 10.1137/0520045.

[13]

S. P. Hastings and J. B. McLeod, Classical Methods in Ordinary Differential Equations. With Applications to Boundary Value Problems, American Mathematical Society, Providence, RI, 2012.

[14] P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.
[15]

H. Hofer, A note on the topological degree at a critical point of mountain pass type, Proc. Amer. Math. Soc., 90 (1984), 309-315. doi: 10.1090/S0002-9939-1984-0727256-0.

[16]

P. Holmes, On a second-order boundary value problem arising in combustion theory, Quart. Appl. Math., 40 (1982/83), 525-538.

[17]

P. Holmes and D. Spence, On a Painlevé-type boundary-value problem, Quart. J. Mech. Appl. Math., 37 (1984), 525-538. doi: 10.1093/qjmam/37.4.525.

[18]

Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $\mathbb{R}^N$ and Séré's non-degeneracy condition, Comm. Partial Differential Equations, 24 (1999), 563-598. doi: 10.1080/03605309908821434.

[19]

G. Karali and C. Sourdis, Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 131-170. doi: 10.1016/j.anihpc.2011.09.005.

[20]

G. Karali and C. Sourdis, Resonance phenomena in a singular perturbation problem in the case of exchange of stabilities, Comm. Partial Differential Equations, 37 (2012), 1620-1667. doi: 10.1080/03605302.2012.681333.

[21]

G. Karali and C. Sourdis, The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit, Arch. Ration. Mech. Anal., 217 (2015), 439-523. doi: 10.1007/s00205-015-0844-3.

[22]

N. KawanoE. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to $Δ u +K(|x|)u^p=0$ in $\mathbb{R}^N$, Funkcial. Ekvac., 36 (1993), 557-579. doi: 10.1619/fesi.52.343.

[23]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1568. doi: 10.1002/cpa.10049.

[24]

P. D. Miller, Applied Asymptotic Analysis, American Mathematical Society, Providence, RI, 2006.

[25]

S. Nakamura, A Remark on eigenvalue splittings for one-dimensional double-well Hamiltonians, Lett. Math. Phys., 11 (1986), 337-340. doi: 10.1007/BF00574159.

[26]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[27]

L. Recke and O. E. Omel'chenko, Boundary layer solutions to problems with infinite-dimensional singular and regular perturbations, J. Differential Equations, 245 (2008), 3806-3822. doi: 10.1016/j.jde.2008.01.017.

[28]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, Vol. 309. Springer-Verlag, Berlin-New York, 1973.

[29]

S. Schecter and C. Sourdis, Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations, J. Dynam. Differential Equations, 22 (2010), 629-655. doi: 10.1007/s10884-010-9171-4.

[30]

C. Sourdis and P. C. Fife, Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces, Adv. Differential Equations, 12 (2007), 623-668.

[31]

S. K. TinN. Kopell and C. K. R. T. Jones, Invariant manifolds and singularly perturbed boundary value problems, SIAM J. Numer. Anal., 31 (1994), 1558-1576. doi: 10.1137/0731081.

[32]

D. L. TurcotteD. A. Spence and H. H. Bau, Multiple solutions for natural convective flows in an internally heated, vertical channel with viscous dissipation and pressure work, Int. J. Heat Mass Transfer, 25 (1982), 699-706. doi: 10.1016/0017-9310(82)90175-2.

[33] W. Walter, Ordinary Differential Equations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.
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