Measure Valued Solutions to a Backward-Forward Heat Equation: A Conference Report

  • M. Slemrod
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 27)


We examine the asymptotic behavior of measure valued solutions to the initial value problem for the nonlinear heat conduction equation
$$ \frac{{\partial u}}{{\partial t}} = \nabla \cdot q(\nabla u),x \in \Omega, t>0 $$
in a bounded domain Ω ⊂R N with boundary condtions of the form
$$ u = 0 on \partial \Omega or q(\nabla u) \cdot n = 0 on \partial \Omega $$


Equilibrium Solution Fundamental Theorem Young Measure Springer Lecture Note Nonlinear Dispersive Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. J. M. Ball (1988), A version of the fundamental theorem for Young measures, to appear Proc. CNRS-NSF Workshop on Continuum Theory of Phase Transitions: Nice, France, January 1988, eds. M. Rascle and D. Serre, Springer Lecture Notes in Mathematics.Google Scholar
  2. C. Dellacherie and P-A. Meyer (1975), Probabilities et Potentiel, Hermann, Paris.Google Scholar
  3. R. J. DiPerna (1983a), Convergence of approximate solutions to conservation laws, Archive for Rational Mechanics and Analysis 82, pp. 27–70.MathSciNetMATHCrossRefGoogle Scholar
  4. R. J. DiPema (1983b), Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Physics 91, pp. 1–30.CrossRefGoogle Scholar
  5. R. J. DiPema (1983c), Generalized solutions to conservation laws, in Systems of Nonlinear Partial Differential Equations, NATO ASI Series, ed. J. M. Ball, D. Reidel.Google Scholar
  6. R. J. DiPerna (1985), Measure-valued solutions to conservation laws, Archive for Rational Analysis and Mechanics 88, pp. 223–270.MathSciNetMATHCrossRefGoogle Scholar
  7. R. J. DiPema and A. J. Majda (1987a), Concentrations and regularizations in weak solutions of the incompressible fluid equations, Comm. Math. Physics 108, pp. 667–689.CrossRefGoogle Scholar
  8. R. J. DiPema and A. J. Majda (1987b), Concentrations in regularizations for 2-D incompressible flow, Comm. Pure and Applied Math. 40, pp. 301–345.CrossRefGoogle Scholar
  9. E. J. MacShane (1947), Integration, Princeton Univ. Press.Google Scholar
  10. J. C. Maxwell (1876), On stresses in rarified gases arising from inequalities of temperature, Phil. Trans. Roy. Soc. London 170, pp. 231–256 = Papers 2, pp. 680–712.Google Scholar
  11. I. P. Natanson (1955), Theory of Functions of a Real Variable, vol. 1, F. Unger Publishing Co., New York.Google Scholar
  12. M. E. Schonbek (1982), Convergence of solutions to nonlinear dispersive equations, Comm. in Partial Differential Equations 7, pp. 959–1000.MathSciNetMATHCrossRefGoogle Scholar
  13. M. Slemrod (1989a), Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, nonmonotone clamping, to appear Proc. Royal Soc. Edinburgh.Google Scholar
  14. M. Slemrod (1989b), Trend to equilibrium in the Becker-Döring cluster equations, to appear Nonlinearity.Google Scholar
  15. M. Slemrod (1989c), The relaxed invariance principle and weakly dissipative infinite dimensional dynamical systems, to appear Proc. Conf. on Mixed Problems, ed. K. Kirschgassner, Springer Lecture Notes.Google Scholar
  16. M. Slemrod (1989d), Dynamics of measured valued solutions to a backwards forwards heat equation, submitted to Dynamics and Differential Equations.Google Scholar
  17. L. Tartar (1979), Compensated compactness and applications to partial differential equations in “Nonlinear Analysis and Mechanics”, Herior-Watt Symposium IV, Pitman Research Notes in Mathematics pp. 136–192.Google Scholar
  18. R. Temam (1988), Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.MATHGoogle Scholar
  19. C. Truesdell (1984), Rational Thermodynamics, Second Edition, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  20. C. Truesdell and W. Noll (1965), The Non-Linear Field Theories of Mechanics, in Encyclopedia of Physics, ed. S. Flugge, Vol. III/3, Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • M. Slemrod
    • 1
  1. 1.Center for Mathematical SciencesUniversity of WisconsinMadisonUSA

Personalised recommendations