4,559 research outputs found
On the Burgers-Poisson Equation
In this paper, we prove the existence and uniqueness of weak entropy
solutions to the Burgers-Poisson equation for initial data in L^1(R).
Additional an Oleinik type estimate is established and some criteria on local
smoothness and wave breaking for weak entropy solutions are provided.Comment: 22 page
Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics
We show existence of a global weak dissipative solution of the Cauchy problem
for the two-component Camassa-Holm (2CH) system on the line with nonvanishing
and distinct spatial asymptotics. The influence from the second component in
the 2CH system on the regularity of the solution, and, in particular, the
consequences for wave breaking, is discussed. Furthermore, the interplay
between dissipative and conservative solutions is treated.Comment: arXiv admin note: text overlap with arXiv:1111.318
On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system
The Camassa-Holm equation and its two-component Camassa-Holm system
generalization both experience wave breaking in finite time. To analyze this,
and to obtain solutions past wave breaking, it is common to reformulate the
original equation given in Eulerian coordinates, into a system of ordinary
differential equations in Lagrangian coordinates. It is of considerable
interest to study the stability of solutions and how this is manifested in
Eulerian and Lagrangian variables. We identify criteria of convergence, such
that convergence in Eulerian coordinates is equivalent to convergence in
Lagrangian coordinates. In addition, we show how one can approximate global
conservative solutions of the scalar Camassa-Holm equation by smooth solutions
of the two-component Camassa-Holm system that do not experience wave breaking
Periodic conservative solutions for the two-component Camassa-Holm system
We construct a global continuous semigroup of weak periodic conservative
solutions to the two-component Camassa-Holm system, and , for
initial data in . It is
necessary to augment the system with an associated energy to identify the
conservative solution. We study the stability of these periodic solutions by
constructing a Lipschitz metric. Moreover, it is proved that if the density
is bounded away from zero, the solution is smooth. Furthermore, it is
shown that given a sequence of initial values for the densities that
tend to zero, then the associated solutions will approach the global
conservative weak solution of the Camassa-Holm equation. Finally it is
established how the characteristics govern the smoothness of the solution.Comment: To appear in Spectral Analysis, Differential Equations and
Mathematical Physics, Proc. Symp. Pure Math., Amer. Math. So
Numerical conservative solutions of the Hunter--Saxton equation
In the article a convergent numerical method for conservative solutions of
the Hunter--Saxton equation is derived. The method is based on piecewise linear
projections, followed by evolution along characteristics where the time step is
chosen in order to prevent wave breaking. Convergence is obtained when the time
step is proportional to the square root of the spatial step size, which is a
milder restriction than the common CFL condition for conservation laws
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