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Navier and Stokes meet the wavelet. (English) Zbl 0795.35080
The author considers the Cauchy problem for the Navier-Stokes equations, i.e. \[ v_ t- \Delta v+ (v\cdot\nabla)v+ \nabla p=0,\;\text{div } v=0 \text{ in } \mathbb{R}^ 3\times (0,T), \qquad v(0)= v_ 0 \text{ in } \mathbb{R}^ 3. \] He constructs local strong solutions by making an ansatz of the form \[ v(x,t)= v(x,0)+ \sum_ \alpha c_ \alpha(t) u_ \alpha(x) \] where \((u_ \alpha)\) denotes a suitable wavelet basis. This gives an infinite set of integral equations for the coefficients \(c_ \alpha(t)\) which is solved using Banach’s fixed point theorem. The underlying Banach space is defined with the help of local \(L^ 2\)-norms.

35Q30 Navier-Stokes equations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: DOI
[1] Battle, G., Federbush, P.: A Note on Divergence-Free Vector Wavelets. Preprint · Zbl 0798.42022
[2] Battle, G., Federbush, P.: Divergence-Free Vector Wavelets. Michigan Mathematical Journal40, 181 (1993) · Zbl 0798.42022 · doi:10.1307/mmj/1029004682
[3] Caffarelli, L., Kohn, R., Nirenberg, L.: Partial Regularity of Suitable Weak Solutions of the Navier-Stokes Equations. Comm. Pure Appl. Math.35, 771 (1982) · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[4] Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago Lectures in Mathematics, Chicago: University of Chicago Press 1988 · Zbl 0687.35071
[5] Federbush, P.: Local Strong Solution of the Navier-Stokes Equations in Terms of Local Estimates. Preprint · Zbl 0162.57905
[6] Foias, C., Temam, R.: Some Analytic and Geometric Properties of the Solutions of the Evolution Navier-Stokes Equations. J. Math. Pures et Appl.58, 339 (1979)
[7] Giga, Y., Miyakawa, T.: Navier-Stokes Flow inR 3 with Measures as Initial Vorticity and Morrey Spaces. Comm. PDE14, 577 (1989) · Zbl 0681.35072 · doi:10.1080/03605308908820621
[8] Scheffer, V.: Hausdorff Measure and the Navier-Stokes Equations. Comm. Math. Phys.55, 97 (1977) · Zbl 0357.35071 · doi:10.1007/BF01626512
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