Advanced Search
Article Contents
Article Contents

Measure valued solutions of sub-linear diffusion equations with a drift term

Abstract Related Papers Cited by
  • In this paper we study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by a strictly increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance.
        Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass $m_{\rm c}$, which can be explicitly characterized in terms of $\beta$ and of the drift term. If the initial mass is less then $m_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the stationary solution has a singular part in which the exceeding mass $m- m_{\rm c}$ is accumulated.
    Mathematics Subject Classification: 35K15, 35A21, 35B40.


    \begin{equation} \\ \end{equation}
  • [1]

    L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'' Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.


    L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.


    N. Ben Abdallah, I. Gamba and G. Toscani, Condensation phenomena in Fokker-Planck equations with a super-linear drift, in preparation, 2012.


    A. Braides, "$\Gamma$-Convergence for Beginners,'' Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.


    R. E. Caflisch and C. D. Levermore, Equilibrium for radiation in a homogeneous plasma, Phys. Fluids, 29 (1986), 748-752.doi: 10.1063/1.865928.


    J. A. Carrillo, S. Lisini, G. Savaré and D. Slepčev, Nonlinear mobility continuity equations and generalized displacement convexity, J. Funct. Anal., 258 (2010), 1273-1309.doi: 10.1016/j.jfa.2009.10.016.


    S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases," Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.


    G. Dal Maso, "An Introduction on $\Gamma$-Convergence,'' Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.


    F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J., 33 (1984), 673-709.doi: 10.1512/iumj.1984.33.33036.


    J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.


    M. Escobedo, M. A. Herrero and J. J. L. Velazquez, A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma, Trans. Amer. Math. Soc., 350 (1998), 3837-3901.doi: 10.1090/S0002-9947-98-02279-X.


    A. Figalli and N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions, J. Math. Pures Appl. (9), 94 (2010), 107-130.doi: 10.1016/j.matpur.2009.11.005.


    R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.doi: 10.1137/S0036141096303359.


    G. Kaniadakis and P. Quarati, Classical model of bosons and fermions, Phys. Rev. E, 49 (1994), 5103-5110.doi: 10.1103/PhysRevE.49.5103.


    A. S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4 (1957), 730-737.


    R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.doi: 10.1006/aima.1997.1634.


    F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.doi: 10.1081/PDE-100002243.


    G. Savaré, Gradient flows and evolution variational inequalities in metric spaces, in preparation, 2012.


    J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.


    C. Villani, "Topics in Optimal Transportation,'' Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.


    C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.

  • 加载中

Article Metrics

HTML views() PDF downloads(99) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint