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A well-posed problem for the exterior Stokes equations in two and three dimensions. (English) Zbl 0731.35078

Let \(\Omega\) be a bounded domain of \(R^ n\) \((n=2\) or \(n=3)\) with boundary \(\Gamma\) and let \(\Omega '\) denote the complement of \({\bar \Omega}\). The authors consider the following exterior problem for the steady-state nonhomogeneous Stokes flow: \[ (*)\quad -\nu \Delta \vec u+\nabla p=\vec f,\quad div \vec u=0\text{ in } \Omega ',\quad \vec u|_{\Gamma}=\vec g, \] with a condition on \(\vec u\) at infinity expressed by \[ \int_{\Omega '}| \vec u|^ 2dx<+\infty,\quad \int_{\Omega '}(1/\omega^ 2)| \vec u|^ 2dx<+\infty. \] Here \(\omega\) is a weight function depending upon the dimension, \(\vec u\) is the velocity, p is the pressure, \(\nu\) is the coefficient of viscosity. Using the weighted Sobolev spaces of Hanouzet (in \(R^ 3)\) and Giroire (in \(R^ 2)\), the authors prove the following result: if \(\Omega \subset R^ n\) has a Lipschitz-continuous boundary \(\Gamma\) that is not necessarily connected, but has no interior connected component and \(\vec f\in (W_ 0^{-1}(\Omega '))^ n\), \(\vec g\in (H^{1/2}(\Omega '))^ n\), then the problem (*) has a unique solution (\(\vec u,p)\in (W^ 1_ 0(\Omega '))^ n\times L^ 2(\Omega ')\), which depends continuously on the data \(\vec f,\vec g\). Moreover, they showed that if the boundary and data are smoother, then so is the solution (\(\vec u,p)\) of the Stokes problem.

MSC:

35Q30 Navier-Stokes equations
35J25 Boundary value problems for second-order elliptic equations
35Dxx Generalized solutions to partial differential equations
35A15 Variational methods applied to PDEs
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[1] C. J. Amick, On Leray’s problem of steady Navier-Stokes flow past a body in the plane, Acta Math. 161 (1988), 71-130. · Zbl 0682.76027 · doi:10.1007/BF02392295
[2] K. I. Babenko, On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Mat. Sbornik 91 (133) (1973), 3-26.
[3] W. Borchers & H. Sohr, On the semigroup of the Stokes operator for exterior domains in L q -spaces, Math. Z. 196 (1987), 415-425. · Zbl 0636.76027 · doi:10.1007/BF01200362
[4] F, Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, R.A.I.R.O., Anal. Numer. R2 (1974), 129-151. · Zbl 0338.90047
[5] I. Babuska, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1973), 179-192. · Zbl 0258.65108 · doi:10.1007/BF01436561
[6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308-340.
[7] R. Finn, On the steady-state solutions of the Navier-Stokes equations III, Acta Math. 105 (1961), 197-244. · Zbl 0126.42203 · doi:10.1007/BF02559590
[8] R. Finn, On the exterior stationary problem for the Navier-Stokes equations and associated perturbation problems, Arch. Rational Mech. Anal. 19 (1965), 363-406. · Zbl 0149.44606 · doi:10.1007/BF00253485
[9] H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 9 (1961), 59-102. · Zbl 0111.38502
[10] V. Girault & P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Comp. Math. 5, Springer-Verlag, Berlin (1986). · Zbl 0585.65077
[11] D. Gilbarg & H. F. Weinberger, Asymptotic properties of Leray’s solution of the stationary two-dimensional Navier-Stokes equations, Russian Math. Surveys 29 (1974), 109-123. · Zbl 0304.35071 · doi:10.1070/RM1974v029n02ABEH003843
[12] G. Guirguis, On the existence, uniqueness and regularity of the exterior Stokes problem in ? 3, Comm. in Partial Diff. Eq. 11 (1986), 567-594. · Zbl 0608.35056 · doi:10.1080/03605308608820437
[13] J. Giroire, Etude de Quelques Problèmes aux Limites Extérieurs et Résolution par Equations Intégrales, Thèse, Université Paris VI (1987).
[14] B. Hanouzet, Espaces de Sobolev avec poids ? application au problème de Dirichlet dans un demi-espace, Rend. Sem. Mat. Univ. Padova 46 (1971), 227-272. · Zbl 0247.35041
[15] J. G. Heywood, On stationary solutions of the Navier-Stokes equations and limits of nonstationary solutions, Arch. Rational Mech. Anal. 37 (1970), 48-60 · Zbl 0194.41402 · doi:10.1007/BF00249501
[16] J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), 639-681. · Zbl 0494.35077 · doi:10.1512/iumj.1980.29.29048
[17] G. G. Hardy, D. E. Littlewood & G. Polya, Inequalities, Cambridge Univ. Press (1959).
[18] C. Johnson & J. C. Nedelec, On the coupling of boundary integrals and finite element methods, Math. of Comp. 35 (1980), 1063-1079. · Zbl 0451.65083 · doi:10.1090/S0025-5718-1980-0583487-9
[19] H. Kozono & H. Sohr, L q -regularity theory of the Stokes equations in exterior domains, (preprint). · Zbl 0770.35055
[20] O. A. Ladyzhenskaya & V. A. Solonnikov, On the solvability of boundary and initial-boundary value problems in regions with non-compact boundaries, Vestnik Leningrad Univ. 13 (1977), 39-47. · Zbl 0377.35060
[21] J. Leray, Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82. · Zbl 0006.16702
[22] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), 193-248. · JFM 60.0726.05 · doi:10.1007/BF02547354
[23] C. M. Ma, On square-summability and uniqueness questions concerning nonstationary Stokes flow in an exterior domain, Arch. Rational Mech. Anal. 6 (1979), 99-112. · Zbl 0412.76026
[24] K. Masuda, On the stability of incompressible viscous fluid motions past objects, J. Math. Soc. Japan 27 (1975), 294-327. · Zbl 0303.76011 · doi:10.2969/jmsj/02720294
[25] J. C. Nedelec, Equations intégrales in Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques (R. Dautray & J. L. Lions, eds.) 2, Ch. XI, Collection C.E.A., Masson, Paris (1985).
[26] A. Sequeira, Couplage entre la Méthode des Eléments Finis et la Méthode des Equations Intégrales. Application au Problème Extérieur de Stokes Stationnaire dans le Plan, Thèse, Université Paris VI (1981).
[27] A. Sequeira, The coupling of boundary integral and finite element methods for the bidimensional exterior steady Stokes problem, Math. Meth. in Appl. Sci. 5 (1983), 356-375. · Zbl 0521.76034 · doi:10.1002/mma.1670050124
[28] A. Sequeira, On the computer implementation of a coupled boundary and finite element method for the bidimensional exterior steady Stokes problem, Math. Meth. in Appl. Sci. 8 (1986), 117-133. · Zbl 0619.76039 · doi:10.1002/mma.1670080109
[29] D. R. Smith, Estimates at infinity for stationary solutions of the Navier-Stokes equations in two dimensions, Arch. Rational Mech. Anal. 20 (1965), 341-372. · Zbl 0149.44701 · doi:10.1007/BF00282357
[30] H. Sohr & W. Varnhorn, On decay properties of the Stokes equations in exterior domains (preprint). · Zbl 0719.35069
[31] M. Specovius-Neugebauer, Exterior Stokes problem and decay at infinity, Math. Meth. in Appl. Sci. 8 (1986), 351-367. · Zbl 0616.76033 · doi:10.1002/mma.1670080124
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