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Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow. (English) Zbl 1036.35082

The logarithmic Sobolev inequality in scale of Lizorkin-Triebel spaces (spaces \(F^s_{p,r}\)) is established. The inequality is applied to the regularity problem of the smooth harmonic heat flow into a sphere, \[ u_t -\Delta u=u| \nabla u| ^2, \qquad u(0,x)=u_0(x), \] where \(u(t,x): \mathbb{R}_+\times \mathbb{R}^n \to S^m\). The main result is the possibility of extendint the solution after \(t=T\) under the condition \[ \int^T | | \nabla u(t)| | ^2_{\text{BMO} (\mathbb{R}^n)}\,dt<\infty . \]

MSC:

35K55 Nonlinear parabolic equations
58E20 Harmonic maps, etc.
58J35 Heat and other parabolic equation methods for PDEs on manifolds
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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