{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T19:46:00Z","timestamp":1775763960635,"version":"3.50.1"},"reference-count":13,"publisher":"MathDoc\/Centre Mersenne","issue":"12","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>In this work we consider a planar stationary flow of an incompressible viscous fluid in a semi-infinite strip governed by the standard Stokes system. We show how this fluid can be stopped at a finite distance from the entrance of the semi-infinite strip by means of a feedback source depending in a sublinear way on the velocity field. This localization effect is proved by reducing the problem to a non-linear biharmonic type one for which the localization of solutions is obtained through the application of an energy method, in the spirit of the monograph by S.N. Antontsev, J.I. D\u00edaz and S.I. Shmarev (Energy Methods for Free Boundary Problems: Applications to Non-Linear PDEs and Fluid Mechanics, Birk\u00e4user, Boston,\u00a02002). Since the presence of the non-linear terms defined by the source is not standard in fluid mechanics literature, we give also some results about the existence and uniqueness of weak solutions for this problem.<\/jats:p>","DOI":"10.1016\/s1631-0721(02)01536-x","type":"journal-article","created":{"date-parts":[[2002,12,27]],"date-time":"2002-12-27T10:52:40Z","timestamp":1040986360000},"page":"797-802","source":"Crossref","is-referenced-by-count":8,"title":["On the confinement of a viscous fluid by means of a feedback external field"],"prefix":"10.5802","volume":"330","author":[{"given":"S.N.","family":"Antontsev","sequence":"first","affiliation":[]},{"given":"J.I.","family":"D\u0131\u0301az","sequence":"additional","affiliation":[]},{"given":"H.B.","family":"de\u00a0Oliveira","sequence":"additional","affiliation":[]}],"member":"3842","published-online":{"date-parts":[[2002,11,14]]},"reference":[{"key":"key2025102017330112281_1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0091-8","author":"Antontsev, S.N.","year":"2002","unstructured":"[1] Antontsev, S.N.; D\u00edaz, J.I.; Shmarev, S.I. 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An Introduction to the Mathematical Theory of the Navier\u2013Stokes Equations: Linearised Steady Problems, Springer, New York, 1994","journal-title":"An Introduction to the Mathematical Theory of the Navier\u2013Stokes Equations: Linearised Steady Problems"},{"key":"key2025102017330112281_4","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/BF00281214","article-title":"Elliptic and parabolic semilinear problems without conditions at infinity","volume":"106","author":"Bernis, F.","year":"1989","unstructured":"[4] Bernis, F. Elliptic and parabolic semilinear problems without conditions at infinity, Arch. Rational Mech. Anal., Volume 106 (1989), pp. 217-241","journal-title":"Arch. Rational Mech. Anal."},{"key":"key2025102017330112281_5","author":"Vrabie, I.I.","year":"1987","unstructured":"[5] Vrabie, I.I. Compactness Methods for Non-Linear Evolutions, Pitman, London, 1987","journal-title":"Compactness Methods for Non-Linear Evolutions"},{"key":"key2025102017330112281_6","first-page":"587","article-title":"Strongly non-linear elliptic boundary value problems","volume":"5","author":"Brezis, H.","year":"1978","unstructured":"[6] Brezis, H.; Browder, F.E. Strongly non-linear elliptic boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV), Volume 5 (1978), pp. 587-603","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV)"},{"key":"key2025102017330112281_7","first-page":"245","article-title":"Some properties of higher order Sobolev spaces","volume":"61","author":"Brezis, H.","year":"1982","unstructured":"[7] Brezis, H.; Browder, F.E. Some properties of higher order Sobolev spaces, J. Math. Pures Appl., Volume 61 (1982), pp. 245-259","journal-title":"J. Math. 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Paris, S\u00e9rie I, Volume 297 (1983) no. 3, pp. 149-152","journal-title":"C. R. Acad. Sci. Paris, S\u00e9rie I"},{"issue":"2","key":"key2025102017330112281_10","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1090\/S0002-9947-1985-0792828-X","article-title":"Local vanishing properties of solutions of elliptic and parabolic quasilinear equations","volume":"290","author":"D\u00edaz, J.I.","year":"1985","unstructured":"[10] D\u00edaz, J.I.; V\u00e9ron, L. Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc., Volume 290 (1985) no. 2, pp. 787-814","journal-title":"Trans. Amer. Math. 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Math., Volume 14 (1988), pp. 319-352","journal-title":"Houston J. Math."},{"key":"key2025102017330112281_13","unstructured":"[13] J.I. D\u00edaz, On the formation of the free boundary for the obstacle and Stefan problems via an energy method, in: L. Ferragut, A. Santos (Eds.), Actas XVII CEDYA\/VII CMA (CD-Rom), S.P. Universidad Salamanca, 2001"}],"container-title":["Comptes Rendus. 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