June  2020, 40(6): 3715-3736. doi: 10.3934/dcds.2020028

A functional approach towards eigenvalue problems associated with incompressible flow

1. 

Institute for Mathemaical Sciences, Renmin University of China, Beijing 100872, China

2. 

Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

*Corresponding author: Shuang Liu

Received  September 2018 Revised  April 2019 Published  October 2019

Fund Project: SL was partially supported by the NSFC grant No. 11571364 and the Outstanding Innovative Talents Cultivation Funded Programs 2018 of Renmin Univertity of China. YL was partially supported by the NSF grants DMS-1411176 and DMS-1853561

We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator $ L_{A} = -\mathrm{div}(a(x)\nabla )+A\mathbf{V}\cdot\nabla +c(x) $ and its adjoint operator for general incompressible flow $ \mathbf{V} $. The functional can be applied to establish the monotonicity of the principal eigenvalue $ \lambda_1(A) $, as a function of the advection amplitude $ A $, for the operator $ L_{A} $ subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [5]. The functional can also be used to prove the monotonicity of the normalized speed $ c^{*}(A)/A $ for general incompressible flow, where $ c^{*}(A) $ is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [3] for steady shear flow.

Citation: Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028
References:
[1]

I. Averill, K. Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), ⅴ+117 pp. doi: 10.1090/memo/1161.  Google Scholar

[2]

J. Bedrossian and M. C. Zelati, Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows, Arch. Rational Mech. Anal., 224 (2017), 1161-1204.  doi: 10.1007/s00205-017-1099-y.  Google Scholar

[3]

H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDE's in Condensed Matter and Reactive Flows, 569 (2002), 11-48.  doi: 10.1007/978-94-010-0307-0_2.  Google Scholar

[4]

H. Berestycki and F. Hamel, Front propagatin in periodic excitable Media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.  Google Scholar

[5]

H. BerestyckiF. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480.  doi: 10.1007/s00220-004-1201-9.  Google Scholar

[6]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP-type problems. Ⅰ. Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213.  doi: 10.4171/JEMS/26.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[8]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[9]

X. F. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operators and their applications, Indiana Univ. Math. J., 61 (2012), 45-80.  doi: 10.1512/iumj.2012.61.4518.  Google Scholar

[10]

P. ConstantinA. KiselevL. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. Math., 168 (2008), 643-674.  doi: 10.4007/annals.2008.168.643.  Google Scholar

[11]

A. DevinatzR. Ellis and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives, Ⅱ, Indiana Univ. Math. J., 23 (1973/74), 991-1011.  doi: 10.1512/iumj.1974.23.23081.  Google Scholar

[12]

A. Devinatz and A. Friedman, Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem, Indiana Univ. Math. J., 27 (1978), 143-157.  doi: 10.1512/iumj.1978.27.27012.  Google Scholar

[13]

M. D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci. U.S.A., 72 (1975), 780-783.  doi: 10.1073/pnas.72.3.780.  Google Scholar

[14]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, Ⅰ, Comm. Pure Appl. Math., 28 (1975), 1-47.  doi: 10.1002/cpa.3160280102.  Google Scholar

[15]

M. D. Donsker and S. R. S. Varadhan, On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math., 29 (1976), 595-621.  doi: 10.1002/cpa.3160290606.  Google Scholar

[16]

A. Fannjiang, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives, Indiana Univ. Math. J., 22 (1973), 1005-1015.   Google Scholar

[17]

A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408.  doi: 10.1137/S0036139992236785.  Google Scholar

[18]

T. GodoyJ. P. Gossez and S. Paczka, A minimax formula for the principal eigenvalues of Dirichlet problems and its applications, Proceedings of the 2006 International Conference in honor of Jacqueline Fleckinger, Electron. J. Differ. Equ. Conf., Texas State Univ. -San Marcos, Dept. Math., San Marcos, TX, 16 (2007), 137-154.   Google Scholar

[19]

T. GodoyJ. P. Gossez and S. Paczka, On the asymptotic behavior of the principal eigenvalues of some elliptic problems, Ann. Mat. Pur. Appl., 189 (2010), 497-521.  doi: 10.1007/s10231-009-0120-y.  Google Scholar

[20]

F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), 355-399.  doi: 10.1016/j.matpur.2007.12.005.  Google Scholar

[21]

F. Hamel and N. Nadirashvili, Extinction versus persistence in strong oscillating flows, Arch. Rational Mech. Anal., 195 (2010), 205-223.  doi: 10.1007/s00205-008-0199-0.  Google Scholar

[22]

F. Hamel and A. Zlatoš, Speed-up of combustion fronts in shear flows, Math. Ann., 356 (2013), 845-867.  doi: 10.1007/s00208-012-0877-y.  Google Scholar

[23]

S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics Preprint, 2005. Google Scholar

[24]

V. HutsonK. Michaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[25]

V. HutsonW. Shen and G.T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679.  doi: 10.1090/S0002-9939-00-05808-1.  Google Scholar

[26]

G. IyerA. NovikovL. Ryzhik and A. Zlatoš, Exit times of diffusions with incompressible drift, SIAM J. Math. Anal., 42 (2009), 2484-2498.  doi: 10.1137/090776895.  Google Scholar

[27]

A. KiselevR. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows, Indiana Univ. Math. J., 57 (2008), 2137-2152.  doi: 10.1512/iumj.2008.57.3349.  Google Scholar

[28]

S. Liu, Y. Lou, R. Peng and M. Zhou, Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator, Proc. Amer. Math. Soc., 2019. doi: 10.1090/proc/14653.  Google Scholar

[29]

T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Physics D, 171 (2002), 107-126.  doi: 10.1016/S0167-2789(02)00587-0.  Google Scholar

[30]

P. MichelS. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[31]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4.  Google Scholar

[32]

G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406.  doi: 10.1137/080743597.  Google Scholar

[33]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, Euro. J. Appl. Math., 22 (2011), 169-185.  doi: 10.1017/S0956792511000027.  Google Scholar

[34]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[35]

J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.  Google Scholar

[36]

J. Nolen and J. Xin, Reaction-diffusion front speeds in spatially-temporally periodic shear flows, Multiscale Model. Simul., 1 (2003), 554-570.  doi: 10.1137/S1540345902420234.  Google Scholar

[37]

R. D. Nussbaum and Y. Pinchover, On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications, Festschrift on the occasion of the 70th birthday of Shmuel Agmon, J. Anal. Math., 59 (1992), 161-177.  doi: 10.1007/BF02790223.  Google Scholar

[38]

M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bull. Amer. Math. Soc., 72 (1966), 251-255.  doi: 10.1090/S0002-9904-1966-11485-4.  Google Scholar

[39]

M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.  Google Scholar

[40]

M. E. Smaily and S. Kirsch, Front speed enhancement by incompressible flows in three or higher dimensions, Arch. Rational Mech. Anal., 213 (2014), 327-354.  doi: 10.1007/s00205-014-0725-1.  Google Scholar

[41]

J. VukadinovicE. DeditsA. C. Poje and T. Schäfer, Averaging and spectral properties for the 2D advection-diffusion equation in the semi-classical limit for vanishing diffusivity, Physics D, 310 (2015), 1-18.  doi: 10.1016/j.physd.2015.07.011.  Google Scholar

[42]

A. D. Wentzell, On the asymptotic behavior of the first eigenvalue of a second order differential operator with small parameter in higher derivatives, Theory Prob. Appl., 20 (1975), 599-602.   Google Scholar

[43]

J. X. Xin, Existence of planar flame fronts in convective-diffusive periodic media., Arch. Ration. Mech. Anal., 121 (1992), 205-233.  doi: 10.1007/BF00410613.  Google Scholar

[44]

A. Zlatoš, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Rational Mech. Anal., 195 (2010), 441-453.  doi: 10.1007/s00205-009-0282-1.  Google Scholar

show all references

References:
[1]

I. Averill, K. Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), ⅴ+117 pp. doi: 10.1090/memo/1161.  Google Scholar

[2]

J. Bedrossian and M. C. Zelati, Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows, Arch. Rational Mech. Anal., 224 (2017), 1161-1204.  doi: 10.1007/s00205-017-1099-y.  Google Scholar

[3]

H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDE's in Condensed Matter and Reactive Flows, 569 (2002), 11-48.  doi: 10.1007/978-94-010-0307-0_2.  Google Scholar

[4]

H. Berestycki and F. Hamel, Front propagatin in periodic excitable Media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.  Google Scholar

[5]

H. BerestyckiF. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480.  doi: 10.1007/s00220-004-1201-9.  Google Scholar

[6]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP-type problems. Ⅰ. Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213.  doi: 10.4171/JEMS/26.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[8]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[9]

X. F. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operators and their applications, Indiana Univ. Math. J., 61 (2012), 45-80.  doi: 10.1512/iumj.2012.61.4518.  Google Scholar

[10]

P. ConstantinA. KiselevL. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. Math., 168 (2008), 643-674.  doi: 10.4007/annals.2008.168.643.  Google Scholar

[11]

A. DevinatzR. Ellis and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives, Ⅱ, Indiana Univ. Math. J., 23 (1973/74), 991-1011.  doi: 10.1512/iumj.1974.23.23081.  Google Scholar

[12]

A. Devinatz and A. Friedman, Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem, Indiana Univ. Math. J., 27 (1978), 143-157.  doi: 10.1512/iumj.1978.27.27012.  Google Scholar

[13]

M. D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci. U.S.A., 72 (1975), 780-783.  doi: 10.1073/pnas.72.3.780.  Google Scholar

[14]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, Ⅰ, Comm. Pure Appl. Math., 28 (1975), 1-47.  doi: 10.1002/cpa.3160280102.  Google Scholar

[15]

M. D. Donsker and S. R. S. Varadhan, On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math., 29 (1976), 595-621.  doi: 10.1002/cpa.3160290606.  Google Scholar

[16]

A. Fannjiang, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives, Indiana Univ. Math. J., 22 (1973), 1005-1015.   Google Scholar

[17]

A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408.  doi: 10.1137/S0036139992236785.  Google Scholar

[18]

T. GodoyJ. P. Gossez and S. Paczka, A minimax formula for the principal eigenvalues of Dirichlet problems and its applications, Proceedings of the 2006 International Conference in honor of Jacqueline Fleckinger, Electron. J. Differ. Equ. Conf., Texas State Univ. -San Marcos, Dept. Math., San Marcos, TX, 16 (2007), 137-154.   Google Scholar

[19]

T. GodoyJ. P. Gossez and S. Paczka, On the asymptotic behavior of the principal eigenvalues of some elliptic problems, Ann. Mat. Pur. Appl., 189 (2010), 497-521.  doi: 10.1007/s10231-009-0120-y.  Google Scholar

[20]

F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), 355-399.  doi: 10.1016/j.matpur.2007.12.005.  Google Scholar

[21]

F. Hamel and N. Nadirashvili, Extinction versus persistence in strong oscillating flows, Arch. Rational Mech. Anal., 195 (2010), 205-223.  doi: 10.1007/s00205-008-0199-0.  Google Scholar

[22]

F. Hamel and A. Zlatoš, Speed-up of combustion fronts in shear flows, Math. Ann., 356 (2013), 845-867.  doi: 10.1007/s00208-012-0877-y.  Google Scholar

[23]

S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics Preprint, 2005. Google Scholar

[24]

V. HutsonK. Michaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[25]

V. HutsonW. Shen and G.T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679.  doi: 10.1090/S0002-9939-00-05808-1.  Google Scholar

[26]

G. IyerA. NovikovL. Ryzhik and A. Zlatoš, Exit times of diffusions with incompressible drift, SIAM J. Math. Anal., 42 (2009), 2484-2498.  doi: 10.1137/090776895.  Google Scholar

[27]

A. KiselevR. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows, Indiana Univ. Math. J., 57 (2008), 2137-2152.  doi: 10.1512/iumj.2008.57.3349.  Google Scholar

[28]

S. Liu, Y. Lou, R. Peng and M. Zhou, Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator, Proc. Amer. Math. Soc., 2019. doi: 10.1090/proc/14653.  Google Scholar

[29]

T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Physics D, 171 (2002), 107-126.  doi: 10.1016/S0167-2789(02)00587-0.  Google Scholar

[30]

P. MichelS. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[31]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4.  Google Scholar

[32]

G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406.  doi: 10.1137/080743597.  Google Scholar

[33]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, Euro. J. Appl. Math., 22 (2011), 169-185.  doi: 10.1017/S0956792511000027.  Google Scholar

[34]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[35]

J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.  Google Scholar

[36]

J. Nolen and J. Xin, Reaction-diffusion front speeds in spatially-temporally periodic shear flows, Multiscale Model. Simul., 1 (2003), 554-570.  doi: 10.1137/S1540345902420234.  Google Scholar

[37]

R. D. Nussbaum and Y. Pinchover, On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications, Festschrift on the occasion of the 70th birthday of Shmuel Agmon, J. Anal. Math., 59 (1992), 161-177.  doi: 10.1007/BF02790223.  Google Scholar

[38]

M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bull. Amer. Math. Soc., 72 (1966), 251-255.  doi: 10.1090/S0002-9904-1966-11485-4.  Google Scholar

[39]

M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.  Google Scholar

[40]

M. E. Smaily and S. Kirsch, Front speed enhancement by incompressible flows in three or higher dimensions, Arch. Rational Mech. Anal., 213 (2014), 327-354.  doi: 10.1007/s00205-014-0725-1.  Google Scholar

[41]

J. VukadinovicE. DeditsA. C. Poje and T. Schäfer, Averaging and spectral properties for the 2D advection-diffusion equation in the semi-classical limit for vanishing diffusivity, Physics D, 310 (2015), 1-18.  doi: 10.1016/j.physd.2015.07.011.  Google Scholar

[42]

A. D. Wentzell, On the asymptotic behavior of the first eigenvalue of a second order differential operator with small parameter in higher derivatives, Theory Prob. Appl., 20 (1975), 599-602.   Google Scholar

[43]

J. X. Xin, Existence of planar flame fronts in convective-diffusive periodic media., Arch. Ration. Mech. Anal., 121 (1992), 205-233.  doi: 10.1007/BF00410613.  Google Scholar

[44]

A. Zlatoš, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Rational Mech. Anal., 195 (2010), 441-453.  doi: 10.1007/s00205-009-0282-1.  Google Scholar

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