Decreasing of the L^1 norm and mass conservation for Porous Medium Equations with advection
DOI:
https://doi.org/10.35819/remat2018v4i2id2959Keywords:
Mass Conservation, Decreasing of the $L^1$ Norm, Porous Medium EquationsAbstract
In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form
u_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u), x \in R^n , t > 0,
where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.Downloads
References
BARRIONUEVO, J. A.; OLIVEIRA, L. S.; ZINGANO, P. R. General asymptotic supnorm estimates for solutions of one-dimensional advection-diffusion equations in heterogeneous media. International Journal of Partial Differential Equations, v. 2014, 8 p., DOI: http://dx.doi.org/10.1155/2014/450417.
BRAZ E SILVA, P.; MELO, W.; ZINGANO, P. R. An asymptotic supnorm estimate for solutions of 1-D systems of convection-diffusion equations. Journal of Differential Equations, v. 258, n. 8, p. 2806-2822, 2015. DOI: 10.1016/j.jde.2014.12.026.
DASKALOPOULOS, P.; KENIG, C. E.Degenerate Diffusions: initial value problems and local regularity theory. Zurich: European Mathematical Society, 2007.
DIBENEDETTO, E. Degenerate Parabolic Equations. New York: Springer, 1993.
DIBENEDETTO, E. On the local behavior of solutions of degenerate parabolic equations with measurable coefficients. Annali della Scuola Normale Superiore di Pisa, v. 13, p. 487-535, 1986.
DIEHL, N. M. L. Contributions to the theory of equations of porous media with advective terms(in Portuguese). PhD Thesis – Programa de Pós-Graduação em Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, September/2015.
FABRIS, L. On the global existence and supnorm estimates for nonnegative solutions of the porous medium equation with arbitrary advection terms (in Portuguese). PhD Thesis – Programa de Pós-Graduação em Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, October/2013.
HU, B. Blow-up Theories for Semilinear Parabolic Equations. Berlin: Springer, 2011.
QUITTNER, P.; SOUPLET, P. Superlinear Parabolic Problems: blow-up, global existence and steady states. Basel: Birkhauser, 2007.
SAMARSKII, A. A.; GALAKTIONOV, V. A.; KURDYUMOV, S.; MIKHAILOV, A. P. Blow-up in Quasilinear Parabolic Equations. Berlin: Walter de Gruyter, 1995.
URBANO, J. M. The Method of Intrinsic Scaling: a systematic approach to regularity for degenerate and singular PDEs. Lecture Notes in Mathematics. v. 1930. New York: Springer, 2008.
VÁZQUEZ, J. L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type. Oxford: Oxford University Press, 2006.
VÁZQUEZ, J. L. The Porous Medium Equation: mathematical theory. Oxford: Oxford University Press, 2007.
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https://orcid.org/0000-0002-0893-7426
