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The Plane Exterior Boundary-Value Problem for Nonhomogeneous Fluids

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Abstract

It is proved that the Stokes problem for nonhomogeneous viscous fluids in an exterior Lipschitz two-dimensional domain has a solution if and only if the data satisfy a suitable compatibility condition. Moreover, it is showed that it is unique in large uniqueness classes.

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Notes

  1. We use standard notation as, e.g., in [3]; \(C_R =\{x\in {\mathbb {R}}^2:r=|{ x}|<R\}\), \(T_R =C_{2R} {\setminus } \overline{C}_R \), \(\Omega _R=\Omega \cap C_R\); \(R_0\) is a fixed positive constant such that \({\mathcal {I}} =\complement C_{R_0}\subset {\Omega }\), where \(\complement C_{R_0}={\mathbb {R}}^2{\setminus }\overline{C}_{R_0}\); \(e_r=r^{-1}{ x}\), for all \(x\ne o\); \(D^{1,q}(\Omega )=\{\phi \in L^1_\mathrm{loc}(\Omega ):\Vert \nabla \phi \Vert _{L^q(\Omega )}<+\infty \}\), \(BMO=\{\phi \in L^1_\mathrm{loc}({\mathbb {R}}^2):\sup _R{\textstyle {1\over R^2}}\int _{C_R}|\phi -{\textstyle {1\over |C_R|}}\int _{C_R}\phi |<+\infty \}\). If S is a vector function space, the symbol \(S_\sigma \) denotes the subset of divergence free elements of S. If f(x) and \(\phi (r)\) are functions defined in \({\mathcal {I}}\), then \(f(x)=o(\phi (r))\) and \(f(x)=O(\phi (r))\) mean respectively that \(\lim _{r\rightarrow +\infty }(f/\phi )=0\) and that \(f/\phi \) is bounded in \({\mathcal {I}}\); c will denote a positive constant whose numerical value in not essential to the purposes of the present paper.

  2. These solutions were called exceptional in [4]. Note that (1.2) is expressed by an ordinary integral since \( T(h,Q)\in L^{q}(\partial \Omega )\) [12], \(q<2+\epsilon \).

  3. Under assumption (1.5) it falls the important case of a mixture of two (or more) immiscible homogeneous viscous fluids.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, viale A. Lincoln 5, 81100, Caserta, Italy

    Remigio Russo & Alfonsina Tartaglione

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  1. Remigio Russo
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  2. Alfonsina Tartaglione
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Correspondence to Alfonsina Tartaglione.

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Russo, R., Tartaglione, A. The Plane Exterior Boundary-Value Problem for Nonhomogeneous Fluids. J. Math. Fluid Mech. 22, 14 (2020). https://doi.org/10.1007/s00021-019-0473-y

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