Local strong solutions of a parabolic system related to
the Boussinesq approximation for buoyancy-driven flow
with viscous heating
| dc.contributor.author | Díaz Díaz, Jesús Ildefonso | |
| dc.contributor.author | Rakotoson, J.M. | |
| dc.contributor.author | Schmidt, P.G. | |
| dc.date.accessioned | 2023-06-20T10:52:17Z | |
| dc.date.available | 2023-06-20T10:52:17Z | |
| dc.date.issued | 2008 | |
| dc.description.abstract | We propose a modification of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven flows of viscous, incompressible fluids. This modification is motivated by unresolved issues regarding the global solvability of the classical system in situations where viscous heating cannot be neglected. A simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. In the present paper, we establish the local-in-time existence and uniqueness of strong solutions for the model problem. The full system of equations and the global-in-time existence of weak solutions will be addressed in forthcoming work. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | DGISGPI (Spain) | |
| dc.description.sponsorship | DGUIC of the CAM and the UCM | |
| dc.description.sponsorship | Programa de Visitantes Distinguidos, Grupo Santander | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/30197 | |
| dc.identifier.issn | 1079-9389 | |
| dc.identifier.officialurl | http://projecteuclid.org/euclid.ade/1355867327 | |
| dc.identifier.relatedurl | http://www.aftabi.com/ | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/51369 | |
| dc.issue.number | 9-10 | |
| dc.journal.title | Advances in Differential Equations | |
| dc.language.iso | eng | |
| dc.page.final | 1000 | |
| dc.page.initial | 977 | |
| dc.publisher | Khayyam Publishing | |
| dc.relation.projectID | MTM2005-03463 | |
| dc.relation.projectID | CCG07-UCM/ESP-2787 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 517.9 | |
| dc.subject.keyword | Boussinesq approximation | |
| dc.subject.keyword | viscous heating | |
| dc.subject.keyword | parabolic system | |
| dc.subject.keyword | strong solutions. | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | Local strong solutions of a parabolic system related to the Boussinesq approximation for buoyancy-driven flow with viscous heating | |
| dc.type | journal article | |
| dc.volume.number | 13 | |
| dcterms.references | Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press. Bruyëre, N. Existence et unicitè de la solution faible-reormailisèe pour un système non linèaire de Bopussinesq. To appear in Conmptes rendues de la Academie des Sciences, Paris. Ser, I. Díaz, J.I., Galiano, G. (1998), Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal di§usion, Topol. Methods Nonlinear Anal. 11, 59-82. Díaz, J.I., Lazzo, M., Schmidt, P.G. (2005), Large solutions for a system of elliptic equations arising from fluid dynamics, SIAM J. Math. Anal. 37, 490-513. Díaz, J.I., Rakotoson, J.M., Schmidt, P.G. A parabolic system involving a quadratic gradient term related to the Boussinesq approximation. Rev. R. Acad. Cien.Serie A Matem, 101, 2007, 113-117. Díaz, J.I., Rakotoson, J.M., Schmidt, P.G., Mathematical issues concerning the Boussinesq approximation for thermally coupled viscous flows, Proc. Appl. Math. Mech. (to appear). Díaz, J.I., Rakotoson, J.M., Schmidt, P.G., Weak solutions of a modified Navier-StokesBoussinesq model for buoyancy-driven flow with viscous heating (in preparation). Díaz, J.I., Vrabie, I.I. (1994), Compactness of the Green operator of nonlinear di§usion equations: Application to Boussinesq type systems in áuid dynamics, Topol. Methods Nonlinear Anal. 4, 399-416 (volume dedicated to Jean Leray) Feireisl, E., Malek, J. (2006), On the Navier-Stokes equations with temperature-dependent transport coefficients, Differ. Equ. Nonlinear Mech., Art. ID 90616,14 pp.(electronic). Hishida, T. (1991), Existence and regularizing properties of solutions for the nonstationary convection problem, Funkcial. Ekvac. 34, 449-474. Kagei, Y. (1993), On weak solutions of nonstationary Boussinesq equations, Differential Integral Equations 6, 587-611. Kagei, Y. (1995), Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function, Hiroshima Math. Journal 25, 251-311. Kagei, Y., Ruuzicka, M., Thäter, G. (2000), Natural convection with dissipative heating, Comm.Math. Phys. 214, 287-313. Laddyzenskaja, O.A., Solonnikov, V.A., Uralíceva, N.N. (1968), Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, A.M.S., Providence, R.I. Lions, P.L. (1996), Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models, Oxford University Press. Mihaljan, J.M. (1962), A rigorous exposition of the Boussinesq approximations applicable to a thin layer of áuid, Astrophys. J. 136, 1126-1133. Morimoto, H. (1992), Nonstationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39, 61-75. Naumann, J. (2006), On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous áuids, Math. Meth. Appl. Sci. 29, 1883-1906. Necas, J., Roubicek, T. (2001), Buoyancy-driven viscous flow with L1-data, Nonlinear Anal. 46,737-755. Rajagopal, K.R., Ruzicka, M., Srinivasa, A.R. (1996), On the Oberbeck-Boussinesq approximation, Math. Models Methods Appl. Sci. 6, 1157-1167. Rakotoson, J.M., Temam, R. (2001), An optimal compactness theorem and application to ellipticparabolic systems, Appl. Math. Lett. 14, 303-306. Temam, R. (1984), Navier-Stokes Equations, North-Holland, Amsterdam | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 | |
| relation.isAuthorOfPublication.latestForDiscovery | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 |
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