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HIGH ORDER GEOMETRIC SMOOTHNESS FOR CONSERVATION LAWS

MARTIN CAMPOS PINTO

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 75252 Paris Cedex 05, France

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ALBERT COHEN

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 75252 Paris Cedex 05, France

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PENCHO PETRUSHEV

Dept. of Math., University of South Carolina, Columbia, SC 29208, USA

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https://doi.org/10.1142/S021989160500035XCited by:0 (Source: Crossref)

Abstract

The smoothness of the solutions of 1D scalar conservation laws is investigated and it is shown that if the initial value has smoothness of order α in L^q with α > 1 and q = 1/α, this smoothness is preserved at any time t > 0 for the graph of the solution viewed as a function in a suitably rotated coordinate system. The precise notion of smoothness is expressed in terms of a scale of Besov spaces which also characterizes the functions that are approximated at rate N^-α in the uniform norm by piecewise polynomials on N adaptive intervals. An important implication of this result is that a properly designed adaptive strategy should approximate the solution at the same rate N^-α in the Hausdorff distance between the graphs.

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References

M. Campos Pintoet al., JHDE 2, 25 (2005). Web of Science, Google Scholar
R. A. DeVore, Acta Numer. 7, 51 (1998). Crossref, Google Scholar
R. A. DeVore and B. J. Lucier, Nonlinear Hyperbolic Problems (Bordeaux, 1988), Lecture Notes in Math. 1402 (Springer, Berlin, 1989) pp. 147–154. Crossref, Google Scholar
R. A. DeVore and B. J. Lucier, Indiana Math. J. 39(2), 413 (1990). Crossref, Web of Science, Google Scholar
R. A. DeVore and V. A. Popov, Function Spaces and Applications, Lecture Notes in Math., 1302 (Springer, Berlin, 1988) pp. 191–205. Crossref, Google Scholar
E. Godlewski and P.-A. Raviart , Numerical Approximation of Hyperbolic Systems of Conservation Laws , Applied Mathematical Sciences 118 ( Springer-Verlag , New York , 1996 ) . Crossref, Google Scholar
D. Kröner , Numerical Schemes for Conservation Laws , Advances in Numerical Mathematics ( Wiley-Teubner , 1997 ) . Google Scholar
S. N. Kruzkov, Math. USSR Sb. 10, 217 (1970). Crossref, Google Scholar
P. D. Lax , Regional Conference Series in Applied Mathematics 11 ( SIAM , Philadelphia , 1973 ) . Google Scholar
R. J. LeVeque , Numerical Methods for Conservation Laws , Lectures in Mathematics ( ETH Zurich. Birkhauser Verlag , Basel , 1990 ) . Crossref, Google Scholar
B. J. Lucier, Math. Comp. 46(173), 59 (1986). Crossref, Web of Science, Google Scholar
P. Petrushev, Function Spaces and Applications, Lecture Notes in Math. 1302 (Springer, Berlin, 1988) pp. 363–377. Crossref, Google Scholar
Bl. Sendov , Hausdorff Approximations ( Kluwer Acad. Pub. , 1990 ) . Crossref, Google Scholar