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Global solutions to the generalized Leray-alpha equation with mixed dissipation terms. (English) Zbl 1334.76035
Summary: Due to the intractability of the Navier-Stokes equation, it is common to study approximating equations. Two of the most common of these are the Leray-\(\alpha\) equation (which replaces the solution \(u\) with \((1 - \alpha^2 \mathcal{L}_1) u\) for a Fourier Multiplier \(\mathcal{L}\)) and the generalized Navier-Stokes equation (which replaces the viscosity term \(\nu \triangle\) with \(\nu \mathcal{L}_2\)). In this paper we consider the combination of these two equations, called the generalized Leray-\(\alpha\) equation. We provide a brief outline of the typical strategies used to solve such equations, and prove, with initial data in a low-regularity \(L^p(\mathbb{R}^n)\) based Sobolev space, the existence of a unique local solution with \(\gamma_1 + \gamma_2 > n / p + 1\). In the \(p = 2\) case, the local solution is extended to a global solution, improving on previously known results.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] D. Barbato, F. Morandin, M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system, arXiv:1407.6734. · Zbl 1309.76053
[2] H. Bessaih, B. Ferrario, The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion, http://arxiv.org/abs/1504.05067 [math.AP]. · Zbl 1354.35087
[3] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem. I, Arch. Ration. Mech. Anal., 16, 269-315, (1964) · Zbl 0126.42301
[4] Gallagher, I.; Planchon, F., On global infinite energy solutions to the Navier-Stokes equations in two dimensions, Arch. Ration. Mech. Anal., 161, 4, 307-337, (2002) · Zbl 1027.35090
[5] Kato, T., The Navier-Stokes equation for an incompressible fluid in \(\mathbf{R}^2\) with a measure as the initial vorticity, Differential Integral Equations, 7, 3-4, 949-966, (1994) · Zbl 0826.35094
[6] Kato, T.; Ponce, G., The Navier-Stokes equation with weak initial data, Int. Math. Res. Not. IMRN, 10, 10, (1994), electronic
[7] Ladyzhenskaya, O. A., (The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, vol. 2, (1969), Gordon and Breach Science Publishers New York), Translated from the Russian by Richard A. Silverman and John Chu · Zbl 0184.52603
[8] Lemarie-Rieusset, P. G., Recent developments in the Navier-Stokes problem, (2002), Chapman and Hall/CRC · Zbl 1034.35093
[9] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 1, 193-248, (1934) · JFM 60.0726.05
[10] Linshiz, Jasmine S.; Titi, Edriss S., Analytical study of certain magnetohydrodynamic-\(\alpha\) models, J. Math. Phys., 48, 6, 28, (2007) · Zbl 1144.81378
[11] Lions, J.-L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod; Gauthier-Villars Paris · Zbl 0189.40603
[12] Pennington, N., Lagrangian averaged Navier-Stokes equation with rough data in Sobolev spaces, J. Math. Anal. Appl., 403, 72-88, (2013) · Zbl 1284.35348
[13] Pennington, N., Local and global low-regularity solutions to generalized Leray-alpha equations, Electron. J. Differential Equations, 2015, 170, 1-24, (2015)
[14] Tao, T., Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data, J. Hyperbolic Differ. Equ., 4, 259-266, (2007) · Zbl 1124.35043
[15] Tao, T., Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2, 3, 361-366, (2009) · Zbl 1190.35177
[16] Taylor, M., Partial differential equations, (1996), Springer-Verlag New York, Inc.
[17] Taylor, M., (Tools for PDE, Mathematical Surveys and Monographs, vol. 81, (2000), American Mathematical Society Providence RI)
[18] Wu, J., Generalized MHD equations, J. Differential Equations, 195, 2, 284-312, (2003) · Zbl 1057.35040
[19] Yamazaki, K., On the global regularity of generalized Leray-alpha type models, Nonlinear Anal. TMA, 75, 2, 503-515, (2012) · Zbl 1233.35064
[20] Yamazaki, K., Logarithmically extended global regularity result of the LANs-alpha MHD system in two dimensions, J. Math. Anal. Appl., 425, 234-248, (2015) · Zbl 1311.35186
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