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Theme for TIFR Centre For Applicable Mathematics, Bangalore

Abstract :  In 1983  K.S. Cheng published  an article  with this title. He built some entropy solutions which do not belong to BV.  Later, in 1994, Lions, Perthame and Tadmor proved a reguralized effect of solutions for nonlinear scalar conservation laws in Sobolev spaces but not in BV.  Fractional BV spaces  $$BV^s$$, 0 < s < 1,  seem to be a natural response to K.S. Cheng on BV.  They are well designed to capture the structure of shock waves. As first applications,  the stability and the optimal smoothing effect in $$BV^s$$ for entropy solutions with some numerical schemes are presented.