Nonuniqueness for the Kinetic Fokker–Planck Equation with Inelastic Boundary Conditions View Full Text


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Article Info

DATE

2019-03

AUTHORS

Hyung Ju Hwang, Juhi Jang, Juan J. L. Velázquez

ABSTRACT

We describe the structure of solutions of the kinetic Fokker–Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient r describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (J Math Kyoto Univ 2:227–235 1963) of the properties of the stochastic process associated to the Fokker–Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if r < rc and unique if rc More... »

PAGES

1-92

References to SciGraph publications

  • 1997-06. A Global Lower Bound for the Fundamental Solution of Kolmogorov-Fokker-Planck Equations in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2004-03. An Invariant Harnack Inequality for a Class of Hypoelliptic Ultraparabolic Equations in MEDITERRANEAN JOURNAL OF MATHEMATICS
  • 2010-03. Global Existence for the Vlasov–Poisson System in Bounded Domains in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2002. Linear and Nonlinear Ultraparabolic Equations of Kolmogorov Type Arising in Diffusion Theory and in Finance in NONLINEAR PROBLEMS IN MATHEMATICAL PHYSICS AND RELATED TOPICS II
  • 2010-11. A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators in POTENTIAL ANALYSIS
  • 2011-12. Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1985. Stochastic Differential Equations, An Introduction with Applications in NONE
  • 2010-12. Global Lp estimates for degenerate Ornstein–Uhlenbeck operators in MATHEMATISCHE ZEITSCHRIFT
  • 1967-12. Hypoelliptic second order differential equations in ACTA MATHEMATICA
  • 2014-10. The Fokker–Planck Equation with Absorbing Boundary Conditions in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1995-09. Singular solutions of the Vlasov-Maxwell system on a half line in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1983. Semigroups of Linear Operators and Applications to Partial Differential Equations in NONE
  • 1976-12. Hypoelliptic differential operators and nilpotent groups in ACTA MATHEMATICA
  • 1963. Linear Partial Differential Operators in NONE
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    http://scigraph.springernature.com/pub.10.1007/s00205-018-1299-0

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    http://dx.doi.org/10.1007/s00205-018-1299-0

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    https://app.dimensions.ai/details/publication/pub.1106341396


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