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The Lagrangian averaged Navier-Stokes equation with rough data in Sobolev spaces. (English) Zbl 1284.35348
Summary: The Lagrangian Averaged Navier-Stokes equation is a recently derived approximation to the Navier-Stokes equation. In this article we prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with low regularity initial data in Sobolev spaces $$W^{s,p}(\mathbb R^n)$$ for $$1<p<\infty$$. For $$L^2$$-based Sobolev spaces, we obtain global existence results. More specifically, we achieve local existence with initial data in the Sobolev space $$H^{n/2p,p}(\mathbb R^n)$$. For initial data in $$H^{3/4,2}(\mathbb R^3)$$, we obtain global existence, improving on previous global existence results, which required data in $$H^{3,2}(\mathbb R^3)$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35A01 Existence problems for PDEs: global existence, local existence, non-existence
##### Keywords:
Navier-Stokes; Lagrangian averaging; global existence
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##### References:
 [1] Bhat, H.; Fetecau, R. C.; Marsden, J. E.; Mohseni, K.; West, M., Lagrangian averaging for compressible fluids, Multiscale Model. Simul., 3, 4, 818-837, (2005), (electronic) · Zbl 1108.76070 [2] Chen, S.; Holm, D. D.; Margolin, L.; Zhang, R., Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133, 1-4, 66-83, (1999), Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998) · Zbl 1194.76080 [3] Cheskidov, Alexey; Holm, Darryl D.; Olson, Eric; Titi, Edriss S., On a Leray-$$\alpha$$ model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 2055, 629-649, (2005) · Zbl 1145.76386 [4] Christ, F.; Weinstein, M., Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100, 87-109, (1991) · Zbl 0743.35067 [5] Kato, T., Strong $$L^p$$-solutions of the Navier Stokes equation in $$\mathbb{R}^m$$, with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073 [6] 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 [7] Kato, T.; Ponce, G., The Navier-Stokes equation with weak initial data, Internat. Math. Res. Notices, 10, (1994) · Zbl 0837.35116 [8] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 7, 891-907, (1988) · Zbl 0671.35066 [9] Kenig, C.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 4, 527-620, (1993) · Zbl 0808.35128 [10] Koch, H.; Tataru, D., Well-posedness for the Navier-Stokes equations, Adv. Math., 157, 1, 22-35, (2001) · Zbl 0972.35084 [11] Ladyzhenskaya, O. A., (The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, vol. 2, (1969), Gordon and Breach Science Publishers New York), Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu [12] Lemarie-Rieusset, P. G., Recent developments in the Navier-Stokes problem, (2002), Chapman and Hall/CRC · Zbl 1034.35093 [13] Linshiz, Jasmine S.; Titi, Edriss S., On the convergence rate of the Euler-$$\alpha$$, an inviscid second-grade complex fluid, model to the Euler equations, J. Stat. Phys., 138, 1-3, 305-332, (2010) · Zbl 1375.35348 [14] Marsden, J.; Ratiu, T.; Shkoller, S., The geometry and analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal., 10, 3, 582-599, (2000) · Zbl 0979.58004 [15] Marsden, J.; Shkoller, S., The anisotropic Lagrangian averaged Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 166, 27-46, (2003) · Zbl 1020.76014 [16] Marsden, J.; Shkoller, S., Global well-posedness for the Lagrangian averaged Navier-Stokes equations on bounded domains, Phil. Trans. R. Soc. Lond., 359, 1449-1468, (2001) · Zbl 1006.35074 [17] Mohseni, K.; Kosović, B.; Shkoller, S.; Marsden, J., Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence, Phys. Fluids, 15, 2, 524-544, (2003) · Zbl 1185.76263 [18] Pennington, N., Recent local and global solutions to the Lagrangian averaged Navier-Stokes equation, Contemp. Math., 581, 217-233, (2012) · Zbl 1320.35263 [19] S. Shkoller, On incompressible averaged Lagrangian hydrodynamics, E-print, http://xyz.lanl.gov/abs/math.AP/9908109, 1999. [20] Shkoller, S., Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom., 55, 145-191, (2000) · Zbl 1044.35061 [21] Taylor, M., Partial differential equations, (1996), Springer-Verlag New York, Inc. [22] Taylor, M., (Tools for PDE, Mathematical Surveys and Monographs, vol. 81, (2000), American Mathematical Society Providence RI)
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