A particle-in-Burgers model: theory and numerics - Laboratoire de ...

Model and motivation Auxiliary steps Results h = 0: coupling h = 0: definition, uniqueness h = 0: numerics, existence The coupled problem
...Frozen particle: understanding the coupling...
Explaination : the Burgers equation with Dirac-at-zero drag term is
equivalent to ∂tu +∂x(u 2 /2) = −λu∂xH.
We introduce Hε ∈ C 1 (R) a non-decreasing function such that Hε(x) = H(x)
when |x| ε. Since we are interested in understanding the behavior of the
solution through the stationary interface{x = 0}, we can study only
stationary solutions. We then obtain the regularized equation for
Uε(x) = u(t, x) in the strip −ε < x < ε:
(U 2 ε/2) ′ (x)+λUε(x)∂xHε(x) = 0.
Proposition (Lagoutière, Seguin, Takahashi ’08)
Independently from the choice of Hε, there exists a solution to the above
ODE with Uε(−ε) = c− and Uε(+ε) = c+ if and only if (c−, c+) ∈ Gλ.
The modelling assumption we make is the following :
the traces γ−u and γ+u at {x = 0} of a solution u of the Burgers equation on
R + ×(R\{0}) are compatible if and only if there exists a solution to above
ODE such that Uε(−ε) = γ−u, Uε(ε) = γ+u.
Thus the germ Gλ is the set of couples(γ−u,γ+u) of possible traces at
{x = 0} (for a.e. t > 0) of the admissible solutions.