A new kind of the solution of degenerate parabolic equation with unbounded convection term

Huashui Zhan

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page Article ID 22, 21 p., electronic only-Article ID 22, 21 p., electronic only
  • ISSN: 2391-5455

Abstract

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A new kind of entropy solution of Cauchy problem of the strong degenerate parabolic equation [...] is introduced. If u0 ∈ L∞(RN), E = {Ei} ∈ (L2(QT))N and divE ∈ L2(QT), by a modified regularization method and choosing the suitable test functions, the BV estimates are got, the existence of the entropy solution is obtained. At last, by Kruzkov bi-variables method, the stability of the solutions is obtained.

How to cite

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Huashui Zhan. "A new kind of the solution of degenerate parabolic equation with unbounded convection term." Open Mathematics 13.1 (2015): Article ID 22, 21 p., electronic only-Article ID 22, 21 p., electronic only. <http://eudml.org/doc/270899>.

@article{HuashuiZhan2015,
abstract = {A new kind of entropy solution of Cauchy problem of the strong degenerate parabolic equation [...] is introduced. If u0 ∈ L∞(RN), E = \{Ei\} ∈ (L2(QT))N and divE ∈ L2(QT), by a modified regularization method and choosing the suitable test functions, the BV estimates are got, the existence of the entropy solution is obtained. At last, by Kruzkov bi-variables method, the stability of the solutions is obtained.},
author = {Huashui Zhan},
journal = {Open Mathematics},
keywords = {Cauchy problem; Degenerate parabolic equation; Entropy solution; Unbounded convection term; anisotropic degenerate parabolic equation; boundary condition; entropy solution},
language = {eng},
number = {1},
pages = {Article ID 22, 21 p., electronic only-Article ID 22, 21 p., electronic only},
title = {A new kind of the solution of degenerate parabolic equation with unbounded convection term},
url = {http://eudml.org/doc/270899},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Huashui Zhan
TI - A new kind of the solution of degenerate parabolic equation with unbounded convection term
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - Article ID 22, 21 p., electronic only
EP - Article ID 22, 21 p., electronic only
AB - A new kind of entropy solution of Cauchy problem of the strong degenerate parabolic equation [...] is introduced. If u0 ∈ L∞(RN), E = {Ei} ∈ (L2(QT))N and divE ∈ L2(QT), by a modified regularization method and choosing the suitable test functions, the BV estimates are got, the existence of the entropy solution is obtained. At last, by Kruzkov bi-variables method, the stability of the solutions is obtained.
LA - eng
KW - Cauchy problem; Degenerate parabolic equation; Entropy solution; Unbounded convection term; anisotropic degenerate parabolic equation; boundary condition; entropy solution
UR - http://eudml.org/doc/270899
ER -

References

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  1. [1] Anzellotti G. and Giaquinta M., Funzioni BV e trace. Rend. Sem. Math. Padova, 1978, 60, 1-21 
  2. [2] Baccardo L., Orsina L. and Porretta A., Some noncoercive parabolic equations with lower order terms, J. Evol.Equ., 2003, 3, 407- 418 Zbl1032.35104
  3. [3] Bendanamane M. and Karlsen K. H., Reharmonized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SLAM J. Math. Annal., 2004, 36(2), 405-422 
  4. [4] Bénilan Ph., Crandall M. G. and Pierre M., Solutions of the porous medium equation under optimal conditions on initial values, Indiana Univ. Math. J., 1984, 33, 51-87 Zbl0552.35045
  5. [5] Brezis H. and Crandall M. G., Uniqueness of solutions of the initial value problem for ut Δ φ (u) = 0, J. Math.Pures et Appl., 1979, 58, 564-587 
  6. [6] Carrillo J., Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal.,1999, 147, 269-361 Zbl0935.35056
  7. [7] Chen G.Q. and Karlsen K.H., Quasilinear anisotropic degenerate parabolic equations with time-space degenerate diffusion coefficients, Comm. Pure. Appl. Anal., 2005, 4(2), 2005, 241-267 [Crossref] Zbl1084.35034
  8. [8] Chen G.Q. and Perthame B., Well-Posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. I. H. Poincare- AN, 2003, 20(4), 645-668 [Crossref] Zbl1031.35077
  9. [9] Cockburn B. and Gripenberg G. , Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equations,1999,151, 231-251 Zbl0921.35017
  10. [10] Deck T. and Ruse S. K., Parabolic differential equations with unbounded coefficients-a generalization of the parametric method, Acta Appl. Math., 2002, 74, 71-91 Zbl1021.35042
  11. [11] Evans L.C., Weak convergence methods for nonlinear partial differential equations, Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics, 74, American Mathematical Society, 1998, 1-66 
  12. [12] Fabrie P. and Gallouet T. , Modeling wells in porous media flows, Math. Models Methods Appl. Sci., 2000, 10, 673-709 Zbl1018.76044
  13. [13] Ishige K. and Murata M. , An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations, Math. Z., 1998, 227, 313-335 Zbl0893.35042
  14. [14] Karlsen K. H. and Ohlberger M., A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations, J. of Math. Ann. Appl., 2002, 275, 439-458 Zbl1072.35545
  15. [15] Kružkov N., First order quasilinear equations in several independent variables, Math. USSR-Sb., 1970, 10, 217-243 
  16. [16] Pinchover Y., On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients, Math. Z., 1996, 223, 569-586 Zbl0869.35010
  17. [17] Vol0pert A.I., BV space and quasilinear equations, Mat. Sb., 1967, 73, 255-302 
  18. [18] Vol0pert A.I. and Hudjaev S.I. , Cauchy’s problem for degenerate second order quasilinear parabolic equations, Mat. Sbornik, 1969, 78(120), 374-396; Engl. Transl.: Math.USSR Sb., 1969, 7(3), 365-387 
  19. [19] Vol0pert A.I. and Hudjaeve S.I., Analysis of class of discontinuous functions and the equations of mathematical physics, Izda. Nauka Moskwa, 1975 (in Russian) 
  20. [20] Wu Zh. and Yin J., Some properties of functions in BVx and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 1989, 5(4), 395-422 
  21. [21] Wu Z. and Zhao J., The first boundary value problem for quasilinear degenerate parabolic equations of second order in several variables, Chin. Ann. of Math., 1983, 4B(3), 319-358 
  22. [22] Wu Z., Zhao J., Yin J. and li H., Nonlinear Diffusion Equations, Word Scientific Publishing, Singapore, 2001. Zbl0997.35001
  23. [23] Zhan H. and Zhao J., The stability of solutions for second order quasilinear degenerate parabolic equations, Acta Math. Sinica, 2007, 50, 615-628 (in chinese) Zbl1142.35487
  24. [24] Zhao J., Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1985, 1(2), 153-165 
  25. [25] Zhao J. and Zhan H., Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations, Science in China Ser. A, 2005, 48, 583-593 Zbl1085.35084

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