Potapov, D. K. Continuous approximations of Gol’dshtik’s model. (English. Russian original) Zbl 1198.35199 Math. Notes 87, No. 2, 244-247 (2010); translation from Mat. Zametki 87, No. 2, 262-266 (2010); erratum Math. Notes 87, No. 3, 453 (2010). Summary: We consider continuous approximations to the Gol’dshtik problem for separated flows in an incompressible fluid. An approximated problem is obtained from the initial problem by small perturbations of the spectral parameter (vorticity) and by approximating the discontinuous nonlinearity continuously in the phase variable. Under certain conditions, using a variational method, we prove the convergence of solutions of the approximating problems to the solution of the original problem. Cited in 1 ReviewCited in 12 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35A35 Theoretical approximation in context of PDEs 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 76D10 Boundary-layer theory, separation and reattachment, higher-order effects Keywords:continuous approximation; nonlinear elliptic differential equation; boundary-value problem; Laplace operator; discontinuous nonlinearity; separated flow PDFBibTeX XMLCite \textit{D. K. Potapov}, Math. Notes 87, No. 2, 244--247 (2010; Zbl 1198.35199); translation from Mat. Zametki 87, No. 2, 262--266 (2010); erratum Math. Notes 87, No. 3, 453 (2010) Full Text: DOI References: [1] D. K. Potapov, “Stability of basic boundary-value problems of elliptic type with a spectral parameter and discontinuous nonlinearity in the coercive case,” Izv. RAEN. Ser. MMMIU 9(1-2), 159-165 (2005). [2] V. N. Pavlenko and D. K. Potapov, “Approximation of boundary-value problems of elliptic type with a spectral parameter and discontinuous nonlinearity,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 4, 49-55 (2005) [Russian Math. (Iz. VUZ), No. 4, 46-52 (2005)]. · Zbl 1220.35051 [3] D. K. Potapov, “Approximation to the Dirichlet problem for a higher-order elliptic equation with a spectral parameter and a discontinuous nonlinearity,” Differentsial’nyeUravneniya 43(7), 1002-1003 (2007) [Differential Equations 43 (7), 1031-1032 (2007)]. [4] M. A. Krasnosel’skii and A. V. Pokrovskii, “Equations with discontinuous nonlinearities,” Dokl. Akad. Nauk SSSR 248(5), 1056-1059 (1979) [SovietMath. Dokl. 20, 1117-1120 (1979)]. [5] M. A. Gol’dshtik, “A mathematical model of separated flows in an incompressible liquid,” Dokl. Akad. Nauk SSSR 147(6), 1310-1313 (1962) [SovietMath. Dokl. 7, 1090-1093 (1963)]. [6] D. K. Potapov, “A mathematical model of separated flows of incompressible liquid,” Izv. RAEN. Ser.MMMIU 8(3-4), 163-170 (2004). [7] D. K. Potapov, “On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities,” Differentsial’nye Uravneniya 44(5), 715-716 (2008) [Differential Equations 44 (5), 737-739 (2008)]. [8] N. Dunford and J. Schwartz, Linear Operators, Vol. 2: Spectral Theory (New York, 1964; Inostr. Lit., Moscow, 1966). [9] V. N. Pavlenko and D. K. Potapov, “Existence of a ray of eigenvalues for equations with discontinuous operators,” Sibirsk. Mat. Zh. 42(4), 911-919 (2001) [SiberianMath. J. 42 (4), 766-773 (2001)]. · Zbl 0986.47049 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.