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A conditional regularity result for \(p\)-harmonic flows. (English) Zbl 1336.35202

Summary: We prove an \(\varepsilon\)-regularity result for a wide class of parabolic systems \[ u_t- \mathrm{div}(|\nabla u|^{p-2}\nabla u) = B(\cdot, u,\nabla u) \] with the right hand side \(B\) growing critically, like \(|\nabla u|^p\). It is assumed a priori that the solution \(u(t,\cdot)\) is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in the elliptic context by T. Rivière and the last named author [Commun. Partial Differ. Equations 30, No. 4, 589–604 (2005; Zbl 1157.35376)].

MSC:

35K65 Degenerate parabolic equations
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35K55 Nonlinear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Citations:

Zbl 1157.35376
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References:

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