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Feller evolution families and parabolic equations with form-bounded vector fields

1407.4861

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      Abstract

      We show that the weak solutions of parabolic equation \(\partial_t u - \Delta u + b(t,x) \cdot \nabla u=0\), \((t,x) \in (0,\infty) \times \mathbb R^d\), \(d \geqslant 3\), for \(b(t,x)\) in a wide class of time-dependent vector fields capturing critical singularities both in time and in spatial variables, constitute a Feller evolution family and, thus, determine a Feller process. Our proof uses a Moser-type iterative procedure and an a priori estimate on the \(L^p\)-norm of the gradient of solution in terms of the \(L^q\)-norm of the gradient of initial function.

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      Journal
      2014-07-17
      2015-09-06

      http://arxiv.org/licenses/nonexclusive-distrib/1.0/

      Custom metadata
      35K10, 60G12
      The argument in the proof of Lemma 4 expanded, reference list updated
      math.AP math.PR
      ScienceOpen disciplines:

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