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
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