Compactness of Solutions for Scalar Viscous Conservation Laws in Noncylindrical Domains

W. Neves

Abstract


We utilize the kinetic formulation approach to study the compactness property for the family {u"}">0, solutions of the initial-boundary value problem for the scalar viscous conservation law u" t + divxf(u") = "xu" in a noncylindrical domain. Considering f in C3 and satisfying the non-degeneracy condition, we prove that u" is compact in L1loc.

References


[1] G.-Q. Chen and H. Frid, Large-time behavior of Entropy Solutions in L1 for Multidimensional Scalar Conservation Laws, Advances in nonlinear partial differential equations and related areas, World Sci. Pub., River Edge, NJ, (1998), 28–44.

R. DiPerna and P.L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729–757.

L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, in “CBMS Regional Conference Series in Mathematics No. 74”, AMS, Providence, 1990.

L.C. Evans and R.F. Gariepy, “Lecture Notes on Measure Theory and Fine Properties of Functions”, CRC Press, Boca Raton, Florida, 1992.

H. Federer, “Geometric Measure Theory”, Springer-Verlag, New York, 1969.

G.B. Folland, “Real Analysis: modern techniques and their applications”, John Willey & Sons, Inc., 1999.

H. Frid, Compacidade Compensada Aplicada as Leis de Conservação, em “19 Colóquio Brasileiro de Matemática”, IMPA.

P.L. Lions, B. Perthame and E. Tadmor, A Kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169–192.

W. Neves, Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains, J. Differential Equations, 192 (2003), 360–395.

B. Perthame, “Kinetic Formulation of Conservation Laws”, Oxford Lecture Series in Mathematics and Its Applications, 2003.

L. Schwartz, “Théorie des Distributions”, (2 volumes), Actualites Scientifiques et Industrielles 1091, 1122, Herman, Paris, 1950-51.

D. Serre, “Systems of Conservation Laws”, Vols. 1–2, Cambridge University Press, Cambridge, 1999.




DOI: https://doi.org/10.5540/tema.2003.04.03.0347

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.


Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

         

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia