×

On stationary solutions and inviscid limits for generalized Constantin-Lax-Majda equation with \(O(1)\) forcing. (English) Zbl 1461.76427

Summary: The generalized Constantin-Lax-Majda (gCLM) equation was introduced to model the competing effects of advection and vortex stretching in hydrodynamics. Recent investigations revealed possible connections with the two-dimensional turbulence. With this connection in mind, we consider the steady problem for the viscous gCLM equations on \(\mathbb{T}:av\omega_x-v_x\omega=\nu\Delta\omega+f\), \(v=\left(-\Delta\right)^{-\frac{1}{2}}\omega\), where \(a\in\mathbb{R}\) is the parameter measuring the relative strength between advection and stretching, \(\nu>0\) is the viscosity constant, and \(f\) is a given \(O(1)\)-forcing independent of \(\nu\). For some range of parameters, we establish existence and uniqueness of stationary solutions. We then numerically investigate the behaviour of solutions in the vanishing viscosity limit, where bifurcations appear, and new solutions emerge. When the parameter \(a\) is away from \([-1/2,1]\), we verify that there is convergence towards smooth stationary solutions for the corresponding inviscid equation. Moreover, we analyse the inviscid limit in the fractionally dissipative case, as well as the behaviour of singular limiting solutions.

MSC:

76R99 Diffusion and convection
76F99 Turbulence
35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] AUTO http://indy.cs.concordia.ca/auto/ · Zbl 0786.68067
[2] Bae H, Chae D and Okamoto H 2017 On the well-posedness of various one-dimensional model equations for fluid motion Nonlinear Anal.160 25-43 · Zbl 1457.76071
[3] Luis C and Alexis V 2010 Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation Ann. Math.171 1903-30 · Zbl 1204.35063
[4] Castro A and Córdoba D 2010 Infinite energy solutions of the surface quasi-geostrophic equation Adv. Math.225 1820-9 · Zbl 1205.35219
[5] Chae D and Lee J 2003 Global well-posedness in the super-critical dissipative quasi-geostrophic equations Commun. Math. Phys.233 297-311 · Zbl 1019.86002
[6] Chemin J-Y 1996 A remark on the inviscid limit for two-dimensional incompressible fluids Commun. Part. Differ. Equ.21 1771-9 · Zbl 0876.35087
[7] Chen J 2019 Singularity formation and global well-posedness for the generalized Constantin-Lax-Majda equation with dissipation arXiv:1908.09385
[8] Chen J, Hou T Y and Huang D 2019 On the finite time blowup of the De Gregorio model for the 3D Euler equation arXiv:1905.06387
[9] Chertkov M, Kolokolov I and Lebedev V 2010 Universal velocity profile for coherent vortices in two-dimensional turbulence Phys. Rev. E 81 015302
[10] Constantin P, Lax P D and Majda A 1985 A simple one-dimensional model for the three-dimensional vorticity equation Commun. Pure Appl. Math.38 715-24 · Zbl 0615.76029
[11] Constantin P 2007 On the Euler equations of incompressible fluids Bull. Am. Math. Soc.44 603-21 · Zbl 1132.76009
[12] Constantin P, Majda A J and Tabak E 1994 Formation of strong fronts in the 2D quasigeostrophic thermal active scalar Nonlinearity7 1495-533 · Zbl 0809.35057
[13] Constantin P, Majda A J and Tabak E G 1994 Singular front formation in a model for quasigeostrophic flow Phys. Fluids6 9-11 · Zbl 0826.76014
[14] Constantin P, Tarfulea A and Vicol V 2015 Long time dynamics of forced critical SQG Commun. Math. Phys.335 93-141 · Zbl 1316.35238
[15] Constantin P and Vicol V 2012 Nonlinear maximum principles for dissipative linear nonlocal operators and applications Geomet. Funct. Anal.22 1289-321 · Zbl 1256.35078
[16] Córdoba A and Córdoba D 2004 A maximum principle applied to quasi-geostrophic equations Commun. Math. Phys.249 511-28 · Zbl 1309.76026
[17] Córdoba A, Córdoba D and Fontelos M 2005 Formation of singularities for a transport equation with nonlocal velocity Ann. Math.162 1377-89 · Zbl 1101.35052
[18] De Gregorio S 1990 On a one-dimensional model for the three-dimensional vorticity equation J. Stat. Phys.59 1251-63 · Zbl 0712.76027
[19] De Gregorio S 1996 A partial differential equation arising in a 1D model for the 3D vorticity equation Math. Methods Appl. Sci.19 1233-55 · Zbl 0860.35101
[20] Weinan E 1999 Aubry-Mather theory and periodic solutions of the forced Burgers equation Commun. Pure Appl. Math.52 811-28 · Zbl 0916.35099
[21] Elgindi T 2020 Finite-time singularity formation for C1,α solutions to the incompressible Euler equations on arXiv:1904.04795
[22] Elgindi T M and Jeong I-J 2020 On the effects of advection and vortex stretching Arch. Ration. Mech. Anal.235 1763-817 · Zbl 1434.35091
[23] Elgindi T M and Jeong I-J 2020 Symmetries and critical phenomena in fluids Commun. Pure Appl. Math.73 257-316 · Zbl 1442.76031
[24] Frisch U 1995 Turbulence (Cambridge: Cambridge University Press) The legacy of A N Kolmogorov
[25] Galdi G P 2011 An Introduction to the Mathematical Theory of the Navier-Stokes Equations(Springer Monographs in Mathematics) (New York: Springer) · Zbl 1245.35002
[26] Gallet B and Young W R 2013 A two-dimensional vortex condensate at high Reynolds number J. Fluid Mech.715 359-88 · Zbl 1284.76108
[27] Govaerts W 2000 Numerical Methods for Bifurcations of Dynamical Equilibria · Zbl 0935.37054
[28] Jauslin H R, Kreiss H O and Moser J 1999 On the forced Burgers equation with periodic boundary conditions in Differential Equations: La Pietra 1996 (Florence)(Proceedings of Symposia in Pure Mathematics vol 65) (Providence, RI: American Mathematical Society) pp 133-53 · Zbl 0930.35156
[29] Jeong I-J and Kim S-C 2019 On the stationary solutions and inviscid limit for the generalized Proudman-Johnson equation with O(1) forcing J. Math. Anal. Appl.472 842-63 · Zbl 1418.34050
[30] Jia H, Stewart S and Sverak V 2019 On the De Gregorio modification of the Constantin-Lax-Majda model Arch. Ration. Mech. Anal.231 1269-304 · Zbl 1408.35152
[31] Joseph D D 1976 Stability of Fluid Motions I(Springer Tracts in Natural Philosophy vol 27) (Berlin: Springer)
[32] Ju N 2005 The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations Commun. Math. Phys.255 161-81 · Zbl 1088.37049
[33] Kim S-C, Miyaji T and Okamoto H 2018 Unimodal solutions of the generalized Constantin-Lax-Majda equation with viscosity Jpn. J. Ind. Appl. Math.35 1065-83 · Zbl 1403.76153
[34] Kim S-C and Okamoto H 2010 Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers Jpn. J. Ind. Appl. Math.27 47-71 · Zbl 1204.76012
[35] Kim S-C and Okamoto H 2013 The generalized Proudman-Johnson equation at large Reynolds numbers IMA J. Appl. Math.78 379-403 · Zbl 1402.76037
[36] Kim S-C and Okamoto H 2014 The generalized Proudman-Johnson equation and its singular perturbation problems Jpn. J. Ind. Appl. Math.31 541-73 · Zbl 1309.35082
[37] Kim S-C and Okamoto H 2015 Unimodal patterns appearing in the Kolmogorov flows at large Reynolds numbers Nonlinearity28 3219-42 · Zbl 1446.76093
[38] Kiselev A, Nazarov F and Volberg A 2007 Global well-posedness for the critical 2D dissipative quasi-geostrophic equation Invent Math.167 445-53 · Zbl 1121.35115
[39] Lei Z, Liu J and Ren X 2020 On the Constantin-Lax-Majda model with convection Comm. Math. Phys.375 765-83 · Zbl 1439.35402
[40] Lopes Filho M C, Mazzucato A L and Nussenzveig Lopes H J 2006 Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence Arch. Ration. Mech. Anal.179 353-87 · Zbl 1138.76362
[41] Marchioro C 1986 An example of absence of turbulence for any Reynolds number Commun. Math. Phys.105 99-106 · Zbl 0607.76052
[42] Masmoudi N 2001 Incompressible, inviscid limit of the compressible Navier-Stokes system Ann. Inst. Henri Poincare Anal. C 18 199-224 · Zbl 0991.35058
[43] Masmoudi N 2007 Remarks about the inviscid limit of the Navier-Stokes system Commun. Math. Phys.270 777-88 · Zbl 1118.35030
[44] Matsumoto T and Sakajo T 2016 One-dimensional hydrodynamic model generating a turbulent cascade Phys. Rev. E 93 053101
[45] Matsumoto T and Sakajo T 2017 Turbulence, cascade and singularity in a generalization of the Constantin-Lax-Majda equation arXiv:1707.05205
[46] Di Nezza E, Palatucci G and Valdinoci E 2012 Hitchhiker’s guide to the fractional Sobolev spaces Bull. Sci. Math.136 521-73 · Zbl 1252.46023
[47] Okamoto H, Sakajo T and Marcus W 2008 On a generalization of the Constantin-Lax-Majda equation Nonlinearity21 2447-61 · Zbl 1221.35300
[48] Okamoto H, Sakajo T and Marcus W 2014 Steady-states and traveling-wave solutions of the generalized Constantin-Lax-Majda equation Discrete Continuous Dyn. Syst.34 3155-70 · Zbl 1292.35237
[49] Okamoto H and Zhu J 2000 Some similarity solutions of the Navier-Stokes equations and related topics Proc. of 1999 Int. Conf. on Nonlinear Analysis (Taipei)vol 4 pp 65-103 · Zbl 0972.35090
[50] Sakajo T 2003 Blow-up solutions of the Constantin-Lax-Majda equation with a generalized viscosity term J. Math. Sci. Univ. Tokyo10 187-207 · Zbl 1030.35006
[51] Schochet S 1986 Explicit solutions of the viscous model vorticity equation Commun. Pure Appl. Math.4 531-7 · Zbl 0623.76012
[52] Swann H S G 1971 The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3 Trans. Am. Math. Soc.157 373-97 · Zbl 0218.76023
[53] Tsang Y-K and Young W R 2009 Forced-dissipative two-dimensional turbulence: a scaling regime controlled by drag Phys. Rev. E 79 045308
[54] Yudovich V I 1965 Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible flow Prikl. Mat. Mekh.29 453-67 · Zbl 0148.22307
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.