Publication: Qualitative properties of solutions of some quasilinear equations related to Bingham fluids
Loading...
Official URL
Full text at PDC
Publication Date
2022-12
Authors
Rita Cirmi, Giuseppa
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Citations
Exportar
Abstract
We consider a quasilinear parabolic equation and its associate stationary problem which correspond to a simplified formulation of a Bingham flow and we mainly study two qualitative properties. The first one concerns with the Absence and, respectively, disappearance in finite time, of the movement. We show that there is a suitable balance between the L1-norm of the forcing datum f∞ and the measure of the spatial domain Ω (essentially saying that the forcing daum must be small enough) such that the corresponding solution u∞(x) of the stationary problem is such that u∞ ≡ 0 a.e. in Ω (even if f∞ ≠ 0). Moreover, if f∞ is also the forcing term of the parabolic problem, and if the above mentioned balance is strict, for any u0 ∈ L(Ω) there exists a finite time Tu0,f∞ > 0 such that the unique solution u(t,x) of the parabolic problem globally stops after Tu0,f∞, in the sense that u(t,x) ≡ 0 a.e. in Ω, for any t ≥ Tu0,f∞. The second property concerns with the Formation of a positively measure “solid region”. We show that if the above balance condition fails (i.e., when the forcing datum is large enough) then the solution u∞(x) of the stationary problem satisfies that u∞ ≠ 0 in Ω and its “solid region” (defined as the set S(u∞) = {x ∈ Ω : ∇u∞(x) = 0}) has a positive measure. Similar results are obtained for the symmetric solutions u(t) of the parabolic problem. In addition the convergence u(t) → u∞ in H10 (Ω), as t → +∞, does not take place in any finite time.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] F. Andreu, V. Caselles, J.I. Díaz and J.M. Mazón, Some Qualitative properties for the Total Variation Flow, J. Funct. Anal. 188 (2002), 516-547.
[2] S.N. Antontsev, J.I. Díaz and S.I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, 48, Birkhäuser, Boston, 2002.
[3] G. Bellettini, V. Caselles, M. Novaga, The Total Variation Flow in RN. J. Differential Equations, 184, 475-525 (2002).
[4] Ph. Benilan, J. I. Díaz. Comparison of solutions of nonlinear evolutions equations with different nonlinear terms. Israel Journal of Mathematics, 42 3 (1982), 241-257.
[5] Ph.Bénilan and C. Picard, Quelques aspects non lineaires du principe du maximum. In: Séminaire de Théorie du Potentiel, Paris, No. 4, F. Hirsch and G. Mokobodzki (eds.), Lecture Notes in Mathematics, 713, Springer, Berlin, 1979, 1-37
[6] E.C. Bingham, An Investigation of the Laws of Plastic Flow U.S. Bureau of Standards Bulletin, 13 (1916) 309-353.
[7] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in: E. Zarantonello (Ed.), Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 101-156.
[8] H. Brézis, Monotone operators, nonlinear semigroups and applications. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 249–255.
[9] H. Brézis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.
[10] T. F. Chan, G. H. Golub and P. Mulet. A nonlinear primal-dual method for total variation-based image restoration. SIAM journal on scientific computing 20 6 (1999), 1964-1977.
[11] L. Chupin, N. Cîndea and G. Lacour, Existence and stopping time for solutions of a class of non newtonian viscous fluids with thixotropic or shear thinning flows, arXiv preprint arXiv:2112.02871 (2021).
[12] E.J. Dean, R. Glowinski, G. Guidoboni, On the numerical simulation of Bingham visco-plastic flow: Old and new results, J. non-Newtonian Fluid Mech. 142, pp. 36-62 (2007).
[13] J.I.Díaz, Anulación de soluciones para operadores acretivos en espacios de Banach. Aplicaciones a ciertos problemas parabólicos no lineales. Rev. Real. Acad. Ciencias Exactas, Físicas y Naturales de Madrid, 74 (1980), 865–880.
[14] J.I.Díaz, Nonlinear PDEs and free boudaries, Pitman, London, 1985. colorblue
[15] J.I.Díaz. Desigualdades de tipo isoperimétrico para problemas de Plateau y capilaridad. Revista de la Academia Canaria de Ciencias, 3 1 (1991), 127-166.
[16] J.I.Díaz, Qualitative Study of Nonlinear Parabolic Equations: an Introduction, Extracta Mathematicae, 16 2 (2001), 303-341.
[17] J.I. Díaz, Simetrización de problemas parabólicas no lineales: aplicación a ecuaciones de reacción-diffusión, Memoria XXVIII de la Real Academia de Ciencias, Madrid, 1991.
[18] J.I. Díaz, Symmetrization of nonlinear elliptic and parabolic problems and applications: a particular overview, in Progress in partial differential equations: elliptic and parabolic problems, C.Bandle et al eds, Pitman Research Notes in Maths, Longman, 1992, 1-16.
[19] J. I. Díaz, R. Glowinski, G. Guidoboni and T. Kim, Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support. Rev. R. Acad. Cien. Serie A. Mat RACSAM 104 1 (2010) 157–200.
[20] J. I. Díaz and F. de Thelin, On a nonlinear parabolic problems arising in some models related to turbulence flows. SIAM Journal of Mathematical Analysis 25 , (1994),1085-1111.
[21] G. Duvaut, J.L. Lions, Les Inéquations en Mécanique et Physique, Dunod, Paris, 1972.
[22] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New-York, 1984.
[23] R. Glowinski, Finite element methods for incompressible viscous flow. In Handbook of Numerical Analysis, Vol. IX, P.G. Ciarlet and J.L. Lions eds., North-Holland, Amsterdam, 2003, 3-1176.
[24] R. Glowinski, P. Le Tallec, Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method, in: T.F. Chan, R. Glowinski, J. Périaux, O.B. Widlund (Eds.), Proceedings of the Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Houston, TX, 1989, SIAM, Philadelphia, USA, 1990, 224-231.
[25] R. Glowinski, J. L. Lions and R. Tremolières, Analyse numérique des Inéquations Variationelles. Vol.2 Applications aux phénomènes stationaires et d’évolution. Dunod, Paris, 1976.
[26] G.H Hardy, J.E. Littlewood and G. Polya, Some simple inequalities satisfied by convex functions. Messenger Math. 58 (1929), 145-152.
[27] P. Harjulehto, P. Hästö, Double phase image restoration, J. Math. Anal. Appl. 501 1 (2021), 123832.
[28] J.W. He and R. Glowinski: Steady Bingham fluid flow in cylindrical pipes: a time dependent approach to the iterative solution, Numerical Algebra with Applications 7 (2000), 381-428.
[29] S. Kamin, S. and L. Véron, Flat core properties associated to the p-Laplace operator, Proceedings of the American Mathematical Society 118 4,(1993), 1079-1085.
[30] G. Mingione and V. Radulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, Journal of Mathematical Analysis and Applications 501 1 (2021), 125-197.
[31] J. Mossino, and J.M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann.Scuola Norm.Pisa, b13, (1986), 51-73.
[32] P. Mossolov and V. Miasnikov, Variational methods in the theory of the fluidity of a viscous-plastic medium, Journal of Mechanics and Applied Mathematics, 73 (1965), 468-492.
[33] L. Orsina and A. C. Ponce, Flat solutions of the 1-Laplacian equation. Rend. Istit. Mat. Univ. Trieste 49 (2017), 41–51.
[34] S. Segura de León and C. M. Webler, Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation, Nonlinear Differential Equations and Applications NoDEA 22 (2015), 1213-1246.
[35] G. Talenti, Best constant in Sobolev inequality, Annali di Matematica Pura ed Applicata 110 (1976), 353-372.
[36] G. Talenti: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. IV, 120 (1977), 159-184.
[37] L.Waite and J. M. Fine, Applied Biofluid Mechanics, McGraw-Hill Professional Publishing, New York, 2007.