**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXVI, 1 (2007)

p. 3 - 10

Finite volume schemes for nonlinear parabolic problems:
another regularization method

R. Eymard, T. Gallouet and R. Herbin

**Abstract**.
On one hand, the existence of a solution to degenerate parabolic equations, without a
nonlinear convection term, can be proven using the results of Alt and Luckhaus, Minty and
Kolmogorov. On the other hand, the proof of uniqueness of an entropy weak solution
to a nonlinear scalar hyperbolic equation,
first provided by Krushkov, has been extended in two directions:
Carrillo has handled the case of degenerate parabolic equations
including a nonlinear convection term, whereas
Di Perna has proven the uniqueness of weaker solutions, namely Young measure entropy solutions.
All of these results are reviewed in the course of a convergence
result for two regularizations of a degenerate parabolic problem including a nonlinear
convective term. The first regularization is classicaly obtained by adding a minimal diffusion,
the second one is given by a finite volume scheme on unstructured meshes. The convergence result is
therefore only based
on *L*^{¥}
(W´(0,*T*))
and
*L*^{2}(0,*T; H*^{1}(W))
estimates, associated with the uniqueness result for a
weaker sense for a solution.

**Keywords**.
Degenerate parabolic equation, entropy weak solution, doubling variable technique,
Young measures, finite volume scheme

**AMS Subject classification.** Primary: 35K65, 35L60, 65M60.

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