Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== If your XXX is L2L^2L2 (the Leray / weak solution framework), you do not run a Picard contraction. Instead you produce a weak solution and then obtain the L∞(0,T;L2)L^\infty(0,T;L^2)L∞(0,T;L2) bound from the energy inequality. ===

Sketch (classical):

Start from a suitable Galerkin approximation unu^nun (finite-dimensional ODEs) with initial data u0u_0u0. For each Galerkin solution multiply the PDE by unu^nun and integrate in space to get

12ddt∥un(t)∥L22+ν∥∇un(t)∥L22=0.\frac12\frac{d}{dt}\|u^n(t)\|_{L^2}^2 + \nu\|\nabla u^n(t)\|_{L^2}^2 = 0.21dtd∥un(t)∥L22+ν∥∇un(t)∥L22=0.
Integrate in time:

∥un(t)∥L22+2ν∫0t∥∇un(s)∥L22 ds=∥u0∥L22.\|u^n(t)\|_{L^2}^2 + 2\nu\int_0^t \|\nabla u^n(s)\|_{L^2}^2\,ds = \|u_0\|_{L^2}^2.∥un(t)∥L22+2ν∫0t∥∇un(s)∥L22ds=∥u0∥L22.
Passing to the weak limit (compactness), one obtains a Leray weak solution uuu satisfying the energy inequality

∥u(t)∥L22+2ν∫0t∥∇u(s)∥L22 ds≤∥u0∥L22,\|u(t)\|_{L^2}^2 + 2\nu\int_0^t \|\nabla u(s)\|_{L^2}^2\,ds \le \|u_0\|_{L^2}^2,∥u(t)∥L22+2ν∫0t∥∇u(s)∥L22ds≤∥u0∥L22,
which implies u∈L∞(0,T;L2)u\in L^\infty(0,T;L^2)u∈L∞(0,T;L2). So the L∞L^\inftyL∞-bound is a consequence of the energy inequality, not an extra hypothesis.

Remarks:
* This route produces weaker solutions (may not be unique and a priori not smooth).
* Upgrading to classical regularity for t>0t>0t>0 then requires further regularity theory / bootstrapping.