High order regularity for solutions of the inviscid burgers equation

  • Ronald A. DeVore
  • Bradley J. Lucier
Hyperbolic P.D.E. Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1402)

Abstract

We discuss a recent Besov space regularity theory for discontinuous, entropy solutions of quasilinear, scalar hyperbolic conservation laws in one space dimension. This theory is very closely related to rates of approximation in L1 by moving grid, finite element methods. In addition, we establish the Besov space regularity of solutions of the inviscid Burgers equation; the new aspect of this study is that no assumption is made about the local variation of the initial data.

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References

  1. [1]
    R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.MATHGoogle Scholar
  2. [2]
    R. A. DeVore and B. J. Lucier, High order regularity for conservation laws, Purdue University Center for Applied Mathematics, Tech. Rep. 85, Aug. 1988.Google Scholar
  3. [3]
    R. A. DeVore and V. A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc., 305 (1988), pp. 397–414.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    —, Interpolation spaces and non-linear approximation, in Function Spaces and Applications, M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin, ed., Springer Lecture Notes in Mathematics, Vol. 1302, Springer-Verlag, New York, 1988, pp. 191–205.CrossRefGoogle Scholar
  5. [5]
    J. Glimm, B. Lindquist, O. McBryan, and L. Padmanabhan, A front tracking reservoir simulator, five-spot validation studies and the water coning problem, in Mathematics of Reservoir Simulation, R. E. Ewing, ed., SIAM, Philadelphia, 1983.Google Scholar
  6. [6]
    P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Regional Conference Series in Applied Mathematics, Vol. 11, SIAM, Philadelphia, 1973.CrossRefGoogle Scholar
  7. [7]
    B. J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp., 46 (1986), pp. 59–69.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    —, Regularity through approximation for scalar conservation laws, SIAM J. Math. Anal., 19 (1988), pp. 763–773.MathSciNetCrossRefGoogle Scholar
  9. [9]
    K. Miller, Alternate modes to control the nodes in the moving finite element method, in Adaptive Computational Methods for Partial Differential Equations, I. Babuska, J. Chandra, J. Flaherty, ed., SIAM, Philadelphia, 1983.Google Scholar
  10. [10]
    P. Petrushev, Direct and converse theorems for best spline approximation with free knots and Besov spaces, C. R. Acad. Bulgare Sci., 39 (1986), pp. 25–28.MathSciNetMATHGoogle Scholar
  11. [11]
    —, Direct and converse theorems for spline and rational approximation and Besov spaces, in Function Spaces and Applications, M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin, ed., Springer Lecture Notes in Mathematics, Vol. 1302, Springer-Verlag, New York, 1988, pp. 363–377.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Ronald A. DeVore
    • 1
  • Bradley J. Lucier
    • 2
  1. 1.Department of MathematicsUniversity of South CarolinaColumbia
  2. 2.Department of MathematicsPurdue UniversityWest Lafayette

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