Nonlinear Hyperbolic Problems pp 147-154 | Cite as

# High order regularity for solutions of the inviscid burgers equation

Hyperbolic P.D.E. Theory

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## 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 *L*^{1} 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.

## Keywords

Besov Space Algebraic Curf Entropy Solution Piecewise Polynomial Springer Lecture Note
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## References

- [1]
- [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]R. A. DeVore and V. A. Popov,
*Interpolation of Besov spaces*, Trans. Amer. Math. Soc., 305 (1988), pp. 397–414.MathSciNetCrossRefzbMATHGoogle Scholar - [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]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]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]B. J. Lucier,
*A moving mesh numerical method for hyperbolic conservation laws*, Math. Comp., 46 (1986), pp. 59–69.MathSciNetCrossRefzbMATHGoogle Scholar - [8]—,
*Regularity through approximation for scalar conservation laws*, SIAM J. Math. Anal., 19 (1988), pp. 763–773.MathSciNetCrossRefGoogle Scholar - [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]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.MathSciNetzbMATHGoogle Scholar - [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

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© Springer-Verlag 1989