Preprint 2000-050

Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws

Abstract: We consider the Cauchy problem for $n\times n$ strictly hyperbolic systems of balance laws

    u_t+f(u)_x=g(x,u),  x \in R, t>0\\
    u(0,.)=u_0 \in L^1 \cap {\bf BV}(R; R^n), \\
    |\lambda_i(u)| \geq c > 0 for all  i\in \{1,\ldots,n\}, \\
    |g(.,u)|+|\nabla_u g(.,u)|\leq \omega \in L^1\cap L^\infty(R),

each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that $\|\omega\|_{L^1(R)}$ and $\|u_o\|_{{\bf BV}(R)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations. Moreover, we give a characterization of the semigroup trajectories in terms of integral estimates.

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Author(s):

Debora Amadori, <Debora.Amadori@mat.unimi.it>

Laurent Gosse, <laurent@mailhost.univaq.it>

Graziano Guerra, <graziano.guerra@unimib.it>

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<graziano.guerra@unimib.it> December 11 2000.