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Communications in Partial Differential Equations Volume 45, 2020 - Issue 12
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Research Article

Existence of weak solutions to a continuity equation with space time nonlocal Darcy law

Luis Caffarellia Department of Mathematics, The University of Texas at Austin, Austin, Texas, USAORCID Iconhttps://orcid.org/0000-0001-9283-7083View further author information
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Maria Gualdania Department of Mathematics, The University of Texas at Austin, Austin, Texas, USACorrespondence[email protected]
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Nicola Zamponib School of Business Informatics and Mathematics, University of Mannheim, Mannheim, GermanyView further author information
Pages 1799-1819 | Received 18 Dec 2019, Accepted 24 Jul 2020, Published online: 08 Sep 2020

References

  • Biler, P., Imbert, C., Karch, G. (2015). The nonlocal porous medium equation: Barenblatt profiles and other weak solutions. Arch. Rational Mech. Anal. 215(2):497–529. DOI: 10.1007/s00205-014-0786-1.
  • Caffarelli, L., Vazquez, J.L. (2016). Regularity of solutions of the fractional porous medium flow with exponent 1/2. St. Petersburg Math. J. 27(3):437–460. DOI: 10.1090/spmj/1397.
  • Caffarelli, L., Vazquez, J.L. (2011). Nonlinear porous medium flow with fractional potential pressure. Arch. Rational Mech. Anal. 202(2):537–565. DOI: 10.1007/s00205-011-0420-4.
  • Serfaty, S., Vazquez, J.L. (2014). A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators. Calc. Var. 49(3–4):1091–1120. DOI: 10.1007/s00526-013-0613-9.
  • Caffarelli, L., Soria, F., Vazquez, J.L. (2013). Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. 15(5):1701–1746. DOI: 10.4171/JEMS/401.
  • Ambrosio, L., Serfaty, S. (2008). A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math. 61(11):1495–1539. DOI: 10.1002/cpa.20223.
  • Giacomin, G., Lebowitz, J. (1997). Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87(1–2):37–61. DOI: 10.1007/BF02181479.
  • Giacomin, G., Lebowitz, J. (1998). Phase segregation dynamics in particle systems with long range interactions. II. Interface motion. SIAM J. Appl. Math. 58(6):1707–1729. DOI: 10.1137/S0036139996313046.
  • Giacomin, G., Lebowitz, J., Marra, R. (2000). Macroscopic evolution of particle systems with short- and long-range interactions. Nonlinearity. 13(6):2143–2162. DOI: 10.1088/0951-7715/13/6/314.
  • Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on Zd. Commun. Math. Phys. 140(3):417–448. DOI: 10.1007/BF02099130.
  • Stan, D., del Teso, F., Vazquez, J. L. (2019). Existence of weak solutions for a general porous medium equation with nonlocal pressure. Arch. Rational Mech. Anal. 233(1):451–496. DOI: 10.1007/s00205-019-01361-0.
  • Lions, P. L., Mas-Gallic, S. (2001). Une méthode particulaire déterministe pour des équations diffusives non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 332(4):369–376. DOI: 10.1016/S0764-4442(00)01795-X.
  • Daus, E., Gualdani, M., Zamponi, N. (2020). Long time behavior and weak-strong uniqueness for a nonlocal porous media equation. J. Differ. Equations. 268(4):1820–1839. DOI: 10.1016/j.jde.2019.09.029.
  • Gasser, I., Marcati, P. (2008). On a generalization of the Div-Curl Lemma. Osaka J. Math. 45:211–214.
  • Feireisl, E., Novotný, A. (2009). Singular Limits in Thermodynamics of Viscous Fluids. Basel: Birkhäuser.
  • Zeidler, E. (1990). Nonlinear Functional Analysis and Its Applications. II /B. New York, NY: Springer-Verlag.
  • Chen, X., Jüngel, A., Liu, J.-G. (2014). A note on Aubin-Lions-Dubinskií lemmas. Acta Appl. Math. 133(1):33–43. DOI: 10.1007/s10440-013-9858-8.

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