On the sign of Colombeau functions and applications to conservation laws

Jiří Jelínek; Dalibor Pražák

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 2, page 245-264
  • ISSN: 0010-2628

Abstract

top
A generalized concept of sign is introduced in the context of Colombeau algebras. It extends the sign of the point-value in the case of sufficiently regular functions. This concept of generalized sign is then used to characterize the entropy condition for discontinuous solutions of scalar conservation laws.

How to cite

top

Jelínek, Jiří, and Pražák, Dalibor. "On the sign of Colombeau functions and applications to conservation laws." Commentationes Mathematicae Universitatis Carolinae 50.2 (2009): 245-264. <http://eudml.org/doc/32496>.

@article{Jelínek2009,
abstract = {A generalized concept of sign is introduced in the context of Colombeau algebras. It extends the sign of the point-value in the case of sufficiently regular functions. This concept of generalized sign is then used to characterize the entropy condition for discontinuous solutions of scalar conservation laws.},
author = {Jelínek, Jiří, Pražák, Dalibor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Colombeau algebra; generalized sign; conservation law; entropy condition; Colombeau algebra; generalized sign; conservation law; entropy condition},
language = {eng},
number = {2},
pages = {245-264},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the sign of Colombeau functions and applications to conservation laws},
url = {http://eudml.org/doc/32496},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Jelínek, Jiří
AU - Pražák, Dalibor
TI - On the sign of Colombeau functions and applications to conservation laws
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 2
SP - 245
EP - 264
AB - A generalized concept of sign is introduced in the context of Colombeau algebras. It extends the sign of the point-value in the case of sufficiently regular functions. This concept of generalized sign is then used to characterize the entropy condition for discontinuous solutions of scalar conservation laws.
LA - eng
KW - Colombeau algebra; generalized sign; conservation law; entropy condition; Colombeau algebra; generalized sign; conservation law; entropy condition
UR - http://eudml.org/doc/32496
ER -

References

top
  1. Colombeau J.-F., 10.1090/S0273-0979-1990-15919-1, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 251--268. Zbl0819.46026MR1028141DOI10.1090/S0273-0979-1990-15919-1
  2. Colombeau J.-F., Elementary introduction to new generalized functions, North-Holland Mathematics Studies 113, Notes on Pure Mathematics 103, North-Holland Publishing Co., Amsterdam, 1985. Zbl0584.46024MR0808961
  3. Dafermos C.M., Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 325, Springer, Berlin, 2000. Zbl1078.35001MR1763936
  4. Danilov V.G., Omel'yanov G.A., Calculation of the singularity dynamics for quadratic nonlinear hyperbolic equations. Example: the Hopf equation, Nonlinear Theory of Generalized Functions (Vienna, 1997), Chapman & Hall/CRC Res. Notes Math. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, 63--74. Zbl0932.35144MR1699862
  5. DiPerna R.J., Lions P.-L., 10.1007/BF01393835, Invent. Math. 98 (1989), no. 3, 511--547. Zbl0696.34049MR1022305DOI10.1007/BF01393835
  6. Lions P.-L., Perthame B., Tadmor E., 10.1090/S0894-0347-1994-1201239-3, J. Amer. Math. Soc. 7 (1994), no. 1, 169--191. Zbl0820.35094MR1201239DOI10.1090/S0894-0347-1994-1201239-3
  7. Lojasiewicz S., Sur la valeur et la limite d'une distribution en un point, Studia Math. 16 (1957), 1--36. Zbl0086.09405MR0087905
  8. Nozari K., Afrouzi G.A., Travelling wave solutions to some PDEs of mathematical physics, Int. J. Math. Math. Sci. (2004), no. 21--24, 1105--1120. Zbl1069.35057MR2085053
  9. Oberguggenberger M., Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series 259, Longman Scientific & Technical, Harlow, 1992. Zbl0818.46036MR1187755
  10. Perthame B., Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications 21, Oxford University Press, Oxford, 2002. Zbl1030.35002MR2064166
  11. Rubio J.E., The global control of shock waves, Nonlinear Theory of Generalized Functions (Vienna, 1997), Chapman & Hall/CRC Res. Notes Math. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 355--367. Zbl0933.35135MR1699875
  12. Rudin W., Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973. Zbl0867.46001MR0365062
  13. Schwartz L., Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX--X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. Zbl0962.46025MR0209834
  14. Shelkovich V.M., 10.1002/mana.200310309, Math. Nachr. 278 (2005), no. 11, 1318--1340. Zbl1115.46035MR2163299DOI10.1002/mana.200310309
  15. Villarreal F., Colombeau's theory and shock wave solutions for systems of PDEs, Electron. J. Differential Equations 2000, no. 21, 17 pp. Zbl0966.46022MR1744084
  16. Ziemer W.P., 10.1007/978-1-4612-1015-3_5, Graduate Texts in Mathematics 120, Springer, New York, 1989. Zbl0692.46022MR1014685DOI10.1007/978-1-4612-1015-3_5

NotesEmbed ?

top

You must be logged in to post comments.