Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems

Philippe Laurençot; Bogdan-Vasile Matioc

Archivum Mathematicum (2023)

  • Volume: 059, Issue: 2, page 201-213
  • ISSN: 0044-8753

Abstract

top
Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.

How to cite

top

Laurençot, Philippe, and Matioc, Bogdan-Vasile. "Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems." Archivum Mathematicum 059.2 (2023): 201-213. <http://eudml.org/doc/298971>.

@article{Laurençot2023,
abstract = {Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.},
author = {Laurençot, Philippe, Matioc, Bogdan-Vasile},
journal = {Archivum Mathematicum},
keywords = {cross diffusion; weak-strong uniqueness; relative entropy},
language = {eng},
number = {2},
pages = {201-213},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems},
url = {http://eudml.org/doc/298971},
volume = {059},
year = {2023},
}

TY - JOUR
AU - Laurençot, Philippe
AU - Matioc, Bogdan-Vasile
TI - Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 059
IS - 2
SP - 201
EP - 213
AB - Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.
LA - eng
KW - cross diffusion; weak-strong uniqueness; relative entropy
UR - http://eudml.org/doc/298971
ER -

References

top
  1. Alt, H.W., Luckhaus, S., 10.1007/BF01176474, Math. Z. 183 (1983), 311–341. (1983) Zbl0497.35049DOI10.1007/BF01176474
  2. Amann, H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. (1993) 
  3. Bénilan, Ph., Crandall, M.G., Pierre, M., 10.1512/iumj.1984.33.33003, Indiana Univ. Math. J. 33 (1984), 51–87. (1984) DOI10.1512/iumj.1984.33.33003
  4. Boyer, F., Fabrie, P., 10.1007/978-1-4614-5975-0, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. (2013) MR2986590DOI10.1007/978-1-4614-5975-0
  5. Brenier, Y., De Lellis, C., Székelyhidi, Jr., L., 10.1007/s00220-011-1267-0, Comm. Math. Phys. 305 (2011), no. 2, 351–361. (2011) MR2805464DOI10.1007/s00220-011-1267-0
  6. Bresch, D., Gisclon, M., Lacroix-Violet, I., 10.1007/s00205-019-01373-w, Arch. Ration. Mech. Anal. 233 (2019), no. 3, 975–1025. (2019) MR3961293DOI10.1007/s00205-019-01373-w
  7. Brézis, H., Crandall, M.G., Uniqueness of solutions of the initial-value problem for u t - Δ ϕ ( u ) = 0 , J. Math. Pures Appl. (9) 58 (1979), 153–163. (1979) 
  8. Chen, X., Jüngel, A., 10.1142/S0218202519500088, Math. Models Methods Appl. Sci. 29 (2019), no. 2, 237–270. (2019) MR3917403DOI10.1142/S0218202519500088
  9. Christoforou, C., Tzavaras, A.E., 10.1007/s00205-017-1212-2, Arch. Ration. Mech. Anal. 229 (2018), no. 1, 1–52. (2018) MR3799089DOI10.1007/s00205-017-1212-2
  10. Escher, J., Matioc, A.-V., Matioc, B.-V., 10.1007/s00021-011-0053-2, J. Math. Fluid Mech. 14 (2012), 267–277. (2012) MR2925108DOI10.1007/s00021-011-0053-2
  11. Feireisl, E., Jin, B.J., Novotný, A., 10.1007/s00021-011-0091-9, J. Math. Fluid Mech. 14 (2012), no. 4, 717–730. (2012) MR2992037DOI10.1007/s00021-011-0091-9
  12. Feireisl, E., Novotný, A., 10.1007/s00205-011-0490-3, Arch. Ration. Mech. Anal. 204 (2012), no. 2, 683–706. (2012) MR2909912DOI10.1007/s00205-011-0490-3
  13. Fischer, J., Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Anal. 159 (2017), 181–207. (2017) MR3659829
  14. Fischer, J., Hensel, S., 10.1007/s00205-019-01486-2, Arch. Ration. Mech. Anal. 236 (2020), no. 2, 967–1087. (2020) MR4072686DOI10.1007/s00205-019-01486-2
  15. Giesselmann, J., Lattanzio, C., Tzavaras, A.E., 10.1007/s00205-016-1063-2, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1427–1484. (2017) MR3594360DOI10.1007/s00205-016-1063-2
  16. Gwiazda, P., Świerczewska-Gwiazda, A., Wiedemann, E., 10.1088/0951-7715/28/11/3873, Nonlinearity 28 (2015), no. 11, 3873–3890. (2015) MR3424896DOI10.1088/0951-7715/28/11/3873
  17. Hopf, K., 10.1142/S0218202522500233, Math. Models Methods Appl. Sci. 32 (2022), 1015–1069. (2022) MR4430363DOI10.1142/S0218202522500233
  18. Huo, X., Jüngel, A., Tzavaras, A.E., 10.1137/21M145210X, SIAM J. Math. Anal. 54 (2022), no. 3, 3215–3252. (2022) MR4429417DOI10.1137/21M145210X
  19. Jüngel, A., Portisch, S., Zurek, A., 10.1016/j.na.2022.112800, Nonlinear Anal. 219 (2022), Paper No. 112800, 1–26. (2022) MR4379345DOI10.1016/j.na.2022.112800
  20. Laurençot, Ph., Matioc, B.-V., Bounded weak solutions to a class of degenerate cross-diffusion systems, arXiv: 2201.06479. 
  21. Laurençot, Ph., Matioc, B.-V., The porous medium equation as a singular limit of the thin film Muskat problem, arXiv:2108.09032, to appear in Asymptot. Anal. 
  22. Laurençot, Ph., Matioc, B.-V., Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals, Trans. Amer. Math. Soc. 375 (2022), no. 8, 5963–5986. (2022) MR4469243
  23. Matioc, B.-V., Walker, Ch., 10.1007/s00605-019-01352-z, Monatsh. Math. 191 (2020), no. 3, 615–634. (2020) MR4064570DOI10.1007/s00605-019-01352-z
  24. Otto, F., 10.1006/jdeq.1996.0155, J. Differential Equations 131 (1996), no. 1, 20–38. (1996) DOI10.1006/jdeq.1996.0155
  25. Pierre, M., Uniqueness of the solutions of u t - Δ ( φ ( u ) ) = 0 with initial datum a measure, Nonlinear Anal. 6 (1982), 175–187. (1982) 
  26. Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. (1978) Zbl0387.46033
  27. Vázquez, J.L., The Porous Medium Equation, Clarendon Press, Oxford, 2007. (2007) MR2286292

NotesEmbed ?

top

You must be logged in to post comments.