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The Rot-Div System in Exterior Domains

Abstract

The goal of this paper is to reconsider the classical elliptic system rot vf, div vg in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L p -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for \({{\bf f} \in \dot B^s_{p,q}(\Omega)}\) for \({-1+\frac 1p < s < \frac 1p}\). As a limit case we prove the result for \({{\bf f} \in \dot B^0_{3,1}(\Omega)}\), provided the boundary is merely in \({B^{2-1/3}_{3,1}}\). The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).

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Correspondence to Piotr B. Mucha.

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Communicated by K. Pileckas

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Mucha, P.B., Pokorný, M. The Rot-Div System in Exterior Domains. J. Math. Fluid Mech. 16, 701–720 (2014). https://doi.org/10.1007/s00021-014-0181-6

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  • DOI: https://doi.org/10.1007/s00021-014-0181-6

Mathematics Subject Classification

  • Primary 35F35
  • Secondary 35J56
  • 35B45

Keywords

  • Rot-div system
  • optimal regularity
  • exterior domain
  • Besov spaces
  • a-priori estimates