DISCONTINUOUS SOLUTIONS FOR
THE DEGASPERIS-PROCESI EQUATION
GIUSEPPE M. COCLITE
(joint work in collaboration with Kenneth H. Karlsen)
In this lecture we consider the nonlinear third order dispersive Degasperis-Procesi equation
ut − utxx + 4uux = 3ux uxx + uuxxx , (t, x) ∈ (0, ∞) × R,
that models shallow water dynamics and its asymptotic accuracy is one order more accurate than
the KdV one. We are interested in the Cauchy problem for this equation, so we augment it with
an initial condition u0 ∈ L1 (R) ∩ BV (R).
Formally, the problem is equivalent to an hyperbolic-elliptic system or to a conservation law
with a nonlocal flux function. We prove the existence and uniqueness of the entropy weak solution,
that is a distributional solution satisfying some additional entropy conditions.
In addition the unique entropy weak solution satisfies the Oleinik type estimate
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ux (t, x) ≤ KT 1 + , 0 < t ≤ T, x ∈ R,
t
for some positive constant KT depending on T and on the total variation of u0 . An implication
is that a shock wave in an entropy weak solution to the Degasperis-Procesi equation is admissible
only if it jumps down in value (like the inviscid Burgers equation).
Finally, we show the existence of a semigroup of solutions associated with the Cauchy problem
for the Degasperis-Procesi equation with the initial condition in L2 ∩ L4 .
(Giuseppe Maria Coclite)
Department of Mathematics
University of Bari
Via E. Orabona 4
70125 Bari, Italy
E-mail address: coclitegm@dm.uniba.it
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Converted from coclite.pdf (PDF).