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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

show all references

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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