A parabolic system involving a quadratic gradient term related to the Boussinesq approximation
| dc.contributor.author | Díaz Díaz, Jesús Ildefonso | |
| dc.contributor.author | Rakotoson, Jean Michel Theresien | |
| dc.contributor.author | Schmidt, Paul G. | |
| dc.date.accessioned | 2023-06-20T09:34:33Z | |
| dc.date.available | 2023-06-20T09:34:33Z | |
| dc.date.issued | 2007 | |
| dc.description.abstract | We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very 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. Based on adequate notions of weak and strong solutions, we establish the global-in-time existence of weak solutions and the uniqueness of strong solutions. | |
| 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.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/15290 | |
| dc.identifier.issn | 1578-7303 | |
| dc.identifier.officialurl | http://www.rac.es/ficheros/doc/00271.pdf | |
| dc.identifier.relatedurl | http://www.rac.es/racsam | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/49944 | |
| dc.issue.number | 1 | |
| dc.journal.title | Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A: Matemáticas | |
| dc.language.iso | eng | |
| dc.page.final | 118 | |
| dc.page.initial | 113 | |
| dc.publisher | Real Academia Ciencias Exactas Físicas Y Naturales | |
| dc.relation.projectID | MTM2005-03463 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.9 | |
| dc.subject.keyword | equations | |
| dc.subject.keyword | fluid | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | A parabolic system involving a quadratic gradient term related to the Boussinesq approximation | |
| dc.type | journal article | |
| dc.volume.number | 101 | |
| dcterms.references | Batchelor, G. K., (1967). An Introduction to Fluid Dynamics, Cambridge University Press. Díaz, J. I. and Galiano, G., (1998). Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11, 59–82. Díaz, J. I., Lazzo, M. and 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. and Schmidt, P. G., Weak solutions of a parabolic system related to the Boussinesq approximation for buoyancy-driven flow with viscous heating (preprint). Díaz, J. I., Rakotoson, J. M. and Schmidt, P. G., Weak solutions of a modified Navier-Stokes-Boussinesq model for buoyancy-driven flow with viscous heating (in preparation). Díaz, J. I. and Vrabie, I. I., (1994). Compactness of the Green operator of nonlinear diffusion equations: Application to Boussinesq type systems in fluid dynamics, Topol. Methods Nonlinear Anal., 4, 399–416 (volume dedicated to Jean Leray). Feireisl, E. and Málek, 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., Růžička, M. Thäter, G., (2000). Natural convection with dissipative heating, Comm. Math. Phys., 214, 287–313. 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 fluid, 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 fluids, Math. Meth. Appl. Sci., 29, 1883–1906. Nečas, J. and Roubíček, T., (2001). Buoyancy-driven viscous flow with L1-data, Nonlinear Anal., 46, 737–755. Rajagopal, K. R., Růžička, M. and Srinivasa, A. R., (1996). On the Oberbeck-Boussinesq approximation, Math. Models Methods Appl. Sci., 6, 1157–1167. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 | |
| relation.isAuthorOfPublication.latestForDiscovery | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 |
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