May  2012, 32(5): 1675-1707. doi: 10.3934/dcds.2012.32.1675

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

1. 

Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia, Italy, Italy, Italy, Italy

Received  November 2010 Revised  July 2011 Published  January 2012

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.
Citation: Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'', Oxford Mathematical Monographs, (2000). Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Lectures in Mathematics ETH Zürich, (2005). Google Scholar

[3]

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

[4]

A. Braides, "$\Gamma$-Convergence for Beginners,'', Oxford Lecture Series in Mathematics and its Applications, 22 (2002). Google Scholar

[5]

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

[6]

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

[7]

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, (1970). Google Scholar

[8]

G. Dal Maso, "An Introduction on $\Gamma$-Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993). Google Scholar

[9]

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

[10]

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

[11]

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. doi: 10.1090/S0002-9947-98-02279-X. Google Scholar

[12]

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. doi: 10.1016/j.matpur.2009.11.005. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

G. Savaré, Gradient flows and evolution variational inequalities in metric spaces,, in preparation, (2012). Google Scholar

[19]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs, (2007). Google Scholar

[20]

C. Villani, "Topics in Optimal Transportation,'', Graduate Studies in Mathematics, 58 (2003). Google Scholar

[21]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009). Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'', Oxford Mathematical Monographs, (2000). Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Lectures in Mathematics ETH Zürich, (2005). Google Scholar

[3]

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

[4]

A. Braides, "$\Gamma$-Convergence for Beginners,'', Oxford Lecture Series in Mathematics and its Applications, 22 (2002). Google Scholar

[5]

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

[6]

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

[7]

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, (1970). Google Scholar

[8]

G. Dal Maso, "An Introduction on $\Gamma$-Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993). Google Scholar

[9]

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

[10]

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

[11]

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. doi: 10.1090/S0002-9947-98-02279-X. Google Scholar

[12]

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. doi: 10.1016/j.matpur.2009.11.005. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

G. Savaré, Gradient flows and evolution variational inequalities in metric spaces,, in preparation, (2012). Google Scholar

[19]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs, (2007). Google Scholar

[20]

C. Villani, "Topics in Optimal Transportation,'', Graduate Studies in Mathematics, 58 (2003). Google Scholar

[21]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009). Google Scholar

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