Complutense University Library

New l(1)-gradient type estimates of solutions to one-dimensional quasilinear parabolic systems

Díaz Díaz, Jesús Ildefonso and Antontsev, S.N. (2010) New l(1)-gradient type estimates of solutions to one-dimensional quasilinear parabolic systems. Communications in contemporary mathematics, 12 (1). pp. 85-106. ISSN 0219-1997

[img] PDF
Restricted to Repository staff only until 31 December 2020.

335kB

Official URL: http://www.worldscinet.com/ccm/ccm.shtml

View download statistics for this eprint

==>>> Export to other formats

Abstract

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L(1)-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton-Jacobi or conservation laws type.


Item Type:Article
Uncontrolled Keywords:laminar hot gas; renormalized solutions; elliptic-equations; colder atmosphere; discharge; existence; space; L(1)-gradient estimates; quasilinear parabolic systems; first order hyperbolic systems
Subjects:Sciences > Mathematics > Differential geometry
Sciences > Mathematics > Differential equations
ID Code:15050
References:

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983) 311–341.

H. Amann, Dynamic theory of quasilinear parabolic equations: II. Reaction-diffusion systems, Differential Integral Equations 3 (1990) 13–75.

K. Ammar and P. Wittbold, Existence of renormalized solution of degenerate elliptic parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 113 (2003) 477–496.

F. Andreu, V. Caselles and J. M. Maz´on, Parabolic Quasilinear Equations Minimizing Linear Growth Functions (Birkhäuser, Bassel, 2004).

S. N. Antontsev and J. I. Díaz, Space and time localization in the flow of two immiscible fluids through a porous medium: Energy methods applied to systems, Nonlinear Anal. 16 (1991) 299–313.

S. N. Antontsev and J. I. Díaz , On the mathematical analysis of the discharge of a laminar hot gas in a colder atmosphere, RACSAM Rev. R. Acad. Cien. Serie A. Mat. 101(1) (2007) 119–124.

S. N. Antontsev and J. I. Díaz , Mathematical treatment of the discharge of a laminar hot gas in a stagnant colder atmosphere, J. Appl. Mech. Tech. Phys. 49(4) (2008) 681–692.

S. N. Antontsev and J. I. Díaz , On thermal and stagnation interfaces generated by the discharge of a laminar hot gas in a stagnant colder atmosphere, in preparation.

S. N. Antontsev and J. I. Díaz , Uniform L1-gradient estimates of solutions solutions to quasilinear parabolic systems in higher dimensions, in preparation.

S. N. Antontsev and J. I. Díaz , On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems, Rev. R. Acad. Cien. Serie A Mat. 103(1) (2009) 201–204.

S. N. Antontsev, J. I. Díaz and A. V. Domanski˘ı, Stability and stabilization of generalized solutions of degenerate problems of two-phase filtration, Dokl. Akad. Nauk 325 (1992) 1151–1155; English translation, Soviet Phys. Dokl. 37(8) (1993) 411–413.

S. N. Antontsev, J. I. Díaz and S. Shmarev, Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics (Bikhäuser, Boston, 2002).

S. N. Antontsev, A. V. Domanskii and V. I. Penkovskii, Filtration in by-well zone of the formation and the well productivity stimulation problems, Lavrentyev Institute of Hydrodynamics, Novosibirsk (1989) (in Russian).

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids (North-Holland Publishing Co., Amsterdam, 1990). Translated from original Russian edition (Nauka, Novosibirsk, 1983).

C. Bardos, A.-Y. Leroux and J.C. Nédelec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979) 1017–1034.

G. Barles, Solutions de viscosit´e des Équations de Hamilton–Jacobi (Springer-Verlag, Paris, 1994).

Ph. Benilan, Équations d’évolution dans un espace de Banach quelconque et application, Thesis, Universite de Paris-Sud (1972).

Ph. Benilan, M. G. Crandall and A. Pazy, Evolution Equations Governed by Accretive Operators, book in preparation.

Ph. Benilan and J.I. Díaz , Pointwise gradient estimates of solutions of one dimensional nonlinear parabolic problems, J. Evol. Equ. 3 (2004) 557–602.

L. Boccardo, D. Giacheti, J. I. Díaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, J. Differential Equations 106(2) (1993) 215–237.

H. Brezis, Analyse fonctionnelle. Théorie et applications (Masson, Paris, 1983).

H. Brezis, On a conjecture of J. Serrin, to appear in Rend. Lincei.

H. Brezis and W. A. Strauss, Semilinear second-order elliptic equations in L1, J. Math. Soc. Japan 25 (1973) 565–590.

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999) 269–361.

J. Carrillo, Conservation laws with discontinuous flux functions and boundary condition, J. Evol. Equ. 3 (2003) 283–301.

M. Chipot, l Goes to Plus Infinity (Birkhäuser, Bassel, 2002).

M. G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972) 108–122.

M. G. Crandall and Th. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 11 (1971) 265–298.

E. DiBenedetto, Degenerate Parabolic Equations (Springer-Verlag, Berlin, 1993).

J. I. Díaz , Nonlinear Partial Differential Equations and Free Boundaries (Pitman, London, 1985).

J. I. Díaz and J. E. Saa, Optimal gradient bounds for some second order quasilinear equations, in Actas del IX CEDYA (Univ. Valladolid, 1987), pp. 147–151.

J. I. Díaz and J. F. Padial, Uniqueness and existence of solutions in the BV(Q) space to a doubly nonlinear parabolic problem, Publ. Mat. 40 (1996) 527–560.

H. Engler, B. Kawohl and S. Luckhaus, Gradient estimates for solutions of parabolic equations and systems, J. Math. Anal. Appl. 147 (1990) 309–329.

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, Boca Raton, FL, 1992).

G. Gagneux and M. Madaune-Tort, Analyse mathèmatique de modèles non linéaires de l’ingénierie pétrolière(Springer, Paris, 1996).

B. H. Gilding, Improved theory for a nonlinear degenerate parabolic equation, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4) 16(2) (1989) 165–224.

E. Hopf, The partial differential equation ut +uux = μuxx, Comm. Pure Appl. Math 3 (1950) 201–230.

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987) 135–176.

S. N. Kruzhkov, Generalized solutions of non linear first order equations and certain quasilinear parabolic equations, Vestnik Moskow Univ. Ser. I Math. Rech. 6 (1964) 67–74.

S. N. Kruzhkov, First-order quasilinear equations in several independent variables, Mat. Sbornik 81 (1970) 228–255.

O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (American Mathematical Society, Providence, R.I., 1967).

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod, Paris, 1969).

P. L. Lions, Generalized Solutions of Hamilton–Jacobi Equations (Pitman, London, 1982).

C. Miranda, Alcune osservazioni sulla maggiorazione in Lυ delle soluzioni deboli delle equazioni ellittiche del secondo ordine, Ann. di Mat. Pura e Appl. (4) 61 (1963) 151–169.

O. A. Oleinik, A. S. Kalashnikov and Y.-L. Chzhou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958) 667–704 (in Russian).

S. Pai, Two-dimensional jet mixing of a compressible fluid, J. Aeronaut. Sci. 16 (1949) 463–469.

S. Pai, Axially symmetrical jet mixing of a compressible fluid, Quart. Appl. Math.10 (1952) 141–148.

S. Pai, Fluid Dynamics of Jets (D. Van Nostrand, Toronto, 1954).

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems (Birkhäuser, Bassel, 2007).

K. Sato, On the generators of non-negative contraction semigroups in Banach lattices, J. Math. Soc. Japan 20 (1968) 431–436.

M. Sánchez-Sanz, A. Sánchez and A. Liñán, Front solutions in high-temperature laminar gas jets, J. Fluid. Mech. 547 (2006) 257–266.

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964) 385–387.

R. P. Sperb, Maximum Principles and Their Applications (Academic Press, New York, 1981).

D. Serre, Syst`emes de lois de conservation, I et II (Diderot Editeur, Arts et Sciences, Paris, 1996).

J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, Berlin, 1983).

G. Stampacchia, Some limit cases of Lp estimates for solutions of second order elliptic equations, Comm. Pure Appl. Math. 16 (1963) 505–510.

J. L. Vázquez, The Porous Medium Equation Mathematical Theory (Oxford University Press, Oxford, 2007).

A. I. Volpert and S. I. Khudayaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics (Nijhoff, Dordrecht, 1985).

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations (World Scientific, New Jersey, 2001).

Deposited On:27 Apr 2012 08:30
Last Modified:06 Feb 2014 10:15

Repository Staff Only: item control page