Cauchy problem for multiscale conservation laws: Application to structured cell populations

Abstract

International audienceIn this paper, we study a vector conservation law that models the growth and selection of ovarian follicles. This work is motivated by a multiscale mathematical model. A two-dimensional conservation law describes the age and maturity structuration of the follicular cell populations. The densities interact through a coupled hyperbolic system between different follicles and cell phases, which results in a vector conservation law and coupling between boundary conditions. The maturity velocity functions possess both a local and a nonlocal character. We prove the existence and uniqueness of the weak solution to Cauchy problem with bounded initial and boundary data

Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

“…On the one hand, existence and uniqueness of the solution in our case is proven for bounded initial conditions, and hence as well the existence of zeroth and first order moments [30]. On the other hand, the same asymptotic behavior as in the smooth velocity case [26], tending towards a monokinetic distribution, is observed in numerical simulations [2,4].…”

“…In the differentiated domain (y > y s ) the cell density is periodic in x. The consequences of the velocity discontinuities on the well-posedness of the model have been tackled in [30], where the existence and uniqueness of the solution is obtained, while a specific numerical treatment has been designed in [2] and [4]. Several strategies have already been explored to reduce the computational costs (parallel computing [1], adaptive mesh refinement [4]).…”

“…The existence and uniqueness of the solution of (2.10) along with initial condition (2.17) and limit conditions (2.12,2.13,2.14, 2.15,2.16,2.18) have been obtained in [30] for the particular choice of velocities and source term that we will consider for the numerical applications in the sequel. In our biological framework, the time dependency of these coefficients also establishes a weak coupling between the N f equations through the first moment in y of their solutions.…”

Adaptive mesh refinement strategy for a non conservative transport problem

“…Formalization in the wavelet framework. The result (15) also applies to the Haar reconstruction (12). This means that the appearance of the spurious noise is due not only to the discontinuities in the solution induced by the localized doubling, but also to the enhancement of the thresholding error.…”

“…105 -F-78153 Le Chesnay, France. 1 corresponding to a doubling transmission condition which must be satisfied by the cell density at the interface between cell cycles (see for instance [12]). Even on a uniform grid, these discontinuous flux conditions are tricky to handle and require a specific scheme in order to achieve high order of accuracy.…”