Renormalized solution for nonlinear degenerate problems in the whole space

Mohamed Maliki[1]; Adama Ouedraogo[2]

  • [1] Department of Mathematics BP 146, Hassan II University Mohammedia (Morocco)
  • [2] Department of Mathematics 03 BP 7021 University of Ouagadougou 03 (Burkina Faso)

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 3, page 597-611
  • ISSN: 0240-2963

Abstract

top
We consider the general degenerate parabolic equation : u t - Δ b ( u ) + d i v F ˜ ( u ) = f in Q = ] 0 , T [ × ℝ N , T > 0 . We suppose that the flux F ˜ is continuous, b is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for L 1 initial data and source term. We establish the uniqueness of this type of solution under a structure condition F ˜ ( r ) = F ( b ( r ) ) and an assumption on the modulus of continuity of b . The novelty of this work is that Ω = ℝ N , u 0 , f ∈ L 1 , b , F ˜ are not Lipschitz functions and the techniques are different from those developed in the previous works.

How to cite

top

Maliki, Mohamed, and Ouedraogo, Adama. "Renormalized solution for nonlinear degenerate problems in the whole space." Annales de la faculté des sciences de Toulouse Mathématiques 17.3 (2008): 597-611. <http://eudml.org/doc/10097>.

@article{Maliki2008,
abstract = {We consider the general degenerate parabolic equation :\[ u\_t - \Delta b(u) + div\ \tilde\{F\}(u) = f \hspace\{28.45274pt\}\hbox\{in\}\ \ Q\ =\ ]0,T[\times \mathbb\{R\}^N , \ \ T&gt;0. \]We suppose that the flux $\tilde\{F\}$ is continuous, $b$ is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for $L^1$ initial data and source term. We establish the uniqueness of this type of solution under a structure condition $\tilde\{F\}(r)=F(b(r))$ and an assumption on the modulus of continuity of $b$. The novelty of this work is that $\Omega =\mathbb\{R\}^\{N\}$, $u_\{0\}$, $f\in L^\{1\}$, $b$, $\tilde\{F\}$ are not Lipschitz functions and the techniques are different from those developed in the previous works.},
affiliation = {Department of Mathematics BP 146, Hassan II University Mohammedia (Morocco); Department of Mathematics 03 BP 7021 University of Ouagadougou 03 (Burkina Faso)},
author = {Maliki, Mohamed, Ouedraogo, Adama},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {non-Lipschitz flux},
language = {eng},
month = {6},
number = {3},
pages = {597-611},
publisher = {Université Paul Sabatier, Toulouse},
title = {Renormalized solution for nonlinear degenerate problems in the whole space},
url = {http://eudml.org/doc/10097},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Maliki, Mohamed
AU - Ouedraogo, Adama
TI - Renormalized solution for nonlinear degenerate problems in the whole space
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 3
SP - 597
EP - 611
AB - We consider the general degenerate parabolic equation :\[ u_t - \Delta b(u) + div\ \tilde{F}(u) = f \hspace{28.45274pt}\hbox{in}\ \ Q\ =\ ]0,T[\times \mathbb{R}^N , \ \ T&gt;0. \]We suppose that the flux $\tilde{F}$ is continuous, $b$ is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for $L^1$ initial data and source term. We establish the uniqueness of this type of solution under a structure condition $\tilde{F}(r)=F(b(r))$ and an assumption on the modulus of continuity of $b$. The novelty of this work is that $\Omega =\mathbb{R}^{N}$, $u_{0}$, $f\in L^{1}$, $b$, $\tilde{F}$ are not Lipschitz functions and the techniques are different from those developed in the previous works.
LA - eng
KW - non-Lipschitz flux
UR - http://eudml.org/doc/10097
ER -

References

top
  1. Ammar (K.), Wittbold (P.).— Existence of renormalized solution of degnerate elliptic-parbolic problems Proc. Royal Soc. Edinb.113A, p. 477-496, (2003). Zbl1077.35103MR1983682
  2. Andreianov (B.P.), Igbida (N.).— Uniqueness for Nonlinear degenerate diffusion-convection problem, to appear in J. Diff. Equat. Zbl1096.35064
  3. Andreianov (B.P.), Bénilan (Ph.), Kruzhkov (S.N.).— L 1 theory of scalar conservation law with continuous flux function. J. Funct. Anal, 171 p. 415-33 (2000). Zbl0944.35048
  4. Alt (H.W.), Luckhauss (S.).— Quasi-linear elliptic-parabolic differential equations, Math.Z., 183, p. 311-341 (1983). Zbl0497.35049
  5. Bénilan (Ph.), Crandall (M.G.), Pazy (A.).— Evolution Equation governed by Accretive operators (book to appear). 
  6. Bénilan (Ph.), Boccardo (L.), Gariepy (B.), Pierre (M.), Vazquez (J.L.).— An L 1 theory of exsitence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scul. Norm. Sup. 22(2) p. 241-273, (1995). Zbl0866.35037MR1354907
  7. Butos (M.C.), Concha (F.), Bürger (R.), Tory (E.M.).— Sedimentation and thickenning : phenomenological foundation and mathematical theory, Kluwer Academic, Dordrecht, (1999). Zbl0936.76001MR1747460
  8. Blanchart (D.), Murat (F.).— Renormalized solutions of nonlinear parabolic problems with L 1 data : existence and uniqueness Proc.royal Soc. Edinb A 127, p. 1137-1152, (1997). Zbl0895.35050MR1489429
  9. Blanchard (D.), Redouane (H.).— Solutions renormalisées d’équations paraboliques à deux non linéaritées. C.R.A.S. Paris 319, p. 831-847, (1994). Zbl0810.35038
  10. Bénilan (Ph.), Gariepy (B.).— Strong solution L 1 of degenerate parabolic equation. J. Diff. Equat., 119, p. 473-502 (1995). Zbl0828.35050MR1340548
  11. Bénilan (Ph.), Kruzhkov (S.N.).— Quasilinear first order equations with continuous non linearities. Russian Acad. Sci. Dokl. Math. Vol 50 N ∘ 3 , p. 391-396 (1995). Zbl0880.35027MR1316937
  12. Bénilan (Ph.), Touré (H.).— Sur l’équation générala u t = ϕ ( u ) x x - ψ ( u ) x + v , C.R. Acad. Sc. Paris, serie 1, 299, 18 (1984). Zbl0586.35016
  13. Bénilan (Ph.), Touré (H.).— Sur l’équation générale u t = a ( . , u , ϕ ( . , u ) x ) x dans L 1 I . Etude du problème stationnaire, in Evolution equations, Lecture Notes Pure and Appl. Math Vol. 168, (1995). 
  14. Bénilan (Ph.), Touré (H.).— Sur l’équation générale u t = a ( . , u , ϕ ( . , u ) x ) x dans L 1 I I . Le problème d’évolution, Ann. Inst. Henri Poincaré, vol. 12, 6, p. 727-761 (1995). Zbl0839.35068
  15. Bénilan (Ph.), Wittbold (P.).— On mild and weak solution of elliptic-Parabolic Problems. Adv. in Diff. Equat. Vol. 1 (6) p. 1053-1072 (1996). Zbl0858.35064MR1409899
  16. Carrillo (J.).— On the uniquness of the solution of the evolution DAM problem, Nonlinear Analysis, Vol 22, N ∘ 5 , p. 573-607 (1999). Zbl0810.76086MR1266545
  17. Carrillo (J.).— Entropy solutions for nonlinear degenerate problems. Arch. Ratio. Mech. Anal. 147, p. 269-361 (1999). Zbl0935.35056MR1709116
  18. Carrillo (J.).— Unicité des solutions du type Kruskov pour des problèmes elliptiques avec des termes de transport non linéaires C. R. Acad. Sc. Paris, t 33, Serie I, N ∘ 5 , (1986). Zbl0623.35030MR854731
  19. Carrillo (J.), Wittbold (P.).— Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems. J. Diff. Equat. 156, p. 93-121 (1999). Zbl0932.35129MR1701806
  20. Chavent (G.), Jaffre (J.).— Mathematical models and finite elements for reservoir simulation, North Holland, Amsterdam, (1986). Zbl0603.76101
  21. Diaz (J.I.), Thelin (F.).— On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM. J. Math. Anal. ; 25, p. 1085-1111 (1994). Zbl0808.35066MR1278892
  22. Gagneux (G.).— Tort (M.M.), Unicité des solutions faibles d’équations de diffusion convection, C. R. Acad. SC. Paris, t 318, Série I, p. 919-924 (1994). Zbl0826.35057MR1278152
  23. Hudjaev (S.N.), Vol’pert (A.I.).— Cauchy’s problem for degenerate second order quasilinear parabolic equation, Math. USSR-Sbornik, Vol 7, N ∘ 3 , p. 365-387 (1969). Zbl0191.11603
  24. Igbida (N.), Urbano (J.M.).— Uniqueness for degenerate problems, NoDEA 10, p. 287-307 (2003). Zbl1024.35054MR1994812
  25. Igbida (N.), Urbano (J.M.).— Continuity results for certain nonlinear parabolic PDEs, Preprint LAMFA, Université de Picadie Jules Vernes. 
  26. Igbida (N.), Wittbold (P.).— Renormalized solution for stephan type problem : Existence and Uniqueness, Preprint LAMFA, Université de Picadie Jules Vernes. 
  27. Kruzhkov (S.N.), Panov (E. Yu.).— Conservative quasilinear first order law in the class of locally sommable functins, Dokl. Akad. Nauk. S.S.S.R. 220, 1 p. 233-26 ; english traduction in soviet Math. Dokl. 16 (1985). 
  28. Landes (R.).— On the existence of weak solutions for quasilinear parabolic initial boundary-value problems.Proc. Royal Soc. Edinb. 89A:217-237, (1981). Zbl0493.35054MR635759
  29. Ladyzenskaja (O.A.), Solonnikov (V.A.) et Ural’ceva (N.N.).— Linear and quasilinear equations of parabolic type, Transl. of Math. Monographs 23 (1968). Zbl0174.15403MR241822
  30. Maliki (M.).— Continuous dependence of the entropy solution of general parabolic equation, Ann. Fac. Sci. Toulouse. Vol.XV,n¡3, p. 589-598 (2006). Zbl1122.35012MR2246415
  31. Bendahmane (M.), Karlsen (K.H.).— Renormalized solutions for quasilinear anisotropic degenerate parabolic equations, Siam J. Marth.Anal. Vol.36, N0.2, pp.405-422, (2004). Zbl1090.35104MR2111783
  32. Maliki (M.), Touré (H.).— Solution généralisée locale d’une équation parabolique quasi linéaire dégénérée du second ordre. Ann. Fac. Sci. Toulouse. Vol. VII 1, (1998) 113-133. Zbl0914.35068MR1658456
  33. Maliki (M.), Touré ( H.).— Dépendence continue de solutions généralisées locales, Ann. Fac. Sci. Toulouse. Vol.X 4, (2001) 701-711. Zbl1029.35027MR1944257
  34. Maliki (M.), Touré ( H.).— Uniqueness of entropy solutions for nonlinear degenerate parabolic problem Journal of Evolution equation 3 (2003), no. 4, 603–622. (Birkhauser). Zbl1052.35106MR2058053
  35. Kruzhkov (S.N.), Panov E.Yu..— Conservative quasilinear first order laws with an infinite domain of dependence on the initial data, Soviet. Math. Dokl. Vol. 42, 2, p. 316-321 (1991). Zbl0789.35039MR1118483
  36. Otto (P.).— L 1 contraction and uniqueness for quasilinear elliptic-parobolic equations J. Diff. Eqa. 131, p. 20-38 (1996). Zbl0862.35078MR1415045
  37. Yin (J.).— On the uniqueness and stability of BV solutions for nonlinear diffusion equations, Comm. Part. Diff. Equat. 15, 12, p. 1671-1683 (1990). Zbl0726.35063MR1080617

NotesEmbed ?

top

You must be logged in to post comments.