Abstract One of the main goals of volcano geodesy is to improve the understanding of how an increase in pressure related to magma accumulation causes ground deformation in order to evaluate volcanic unrest. The inversion methods used for this purpose rely on a parametrization of the shape of the crustal volume in which pressure changes due to magma inflow/outflow (the magma domain), to search for the optimal parameters that minimize the difference between model predicted and measured ground displacements. However, these methods assume a predefined shape of the magma domain, which limits their applicability. Here, we propose a new shape optimization framework that can invert these sources without such prior, formulating a reconstruction problem to infer the complete shape of the magma domain. First, we validate this approach using a synthetic test case and then apply it to observations of the Svartsengi volcanic system in Iceland.
The aim of this article is to prove the existence of a new class of solutions of 1D cubic NLS with an initial data related to a sum of Dirac masses, of critical regularity $F(L^\infty)$, and belonging to $\dot H^s$ for any $s<-1/2$. This problem is motivated by the lack of result for critical regularity initial condition, and also by the study of the vortex filaments dynamics approximated by the binormal flow. Our result is based on a scattering approach, after performing a pseudo-conformal transformation, and on fine estimations of oscillatory integrals.
We consider the flow of a generalized Newtonian fluid through a thin porous medium of thickness epsilon, perforated by periodically distributed solid cylinders of size epsilon. We assume that the fluid is described by the 3D incompressible Stokes system, with a non-linear viscosity following the Carreau law of flow index 1 < r < +infinity, and scaled by a factor epsilon gamma, where gamma is an element of R. Generalizing (Anguiano M.: et al. Q. J. Mech. Math., 75(1), 1-27 (2022)), where the particular case r < 2 and gamma = 1 was addressed, we perform a new and complete study on the asymptotic behavior of the fluid as epsilon goes to zero. Depending on gamma and the flow index r, using homogenization techniques, we derive and rigorously justify different effective linear and non-linear lower-dimensional Darcy's laws. Finally, using a finite element method, we study numerically the influence of the rheological parameters of the fluid and of the shape of the solid obstacles on the behavior of the effective systems.
We consider the initial boundary value problem for the Einstein vacuum equations in the maximal gauge, or more generally, in a gauge where the mean curvature of a timelike foliation is fixed near the boundary. We prove the existence of solutions such that the normal to the boundary is tangent to the time slices, the lapse of the induced time coordinate on the boundary is fixed and the main geometric boundary conditions are given by the 1-parameter family of Riemannian conformal metrics on each two-dimensional section. As in the local existence theory of Christodoulou-Klainerman for the Einstein vacuum equations in the maximal gauge, we use as a reduced system the wave equations satisfied by the components of the second fundamental form of the the time foliation. The main difficulty lies in completing the above set of boundary conditions such that the reduced system is well-posed, but still allows for the recovery of the Einstein equations. We solve this problem by imposing the momentum constraint equations on the boundary, suitably modified by quantities vanishing in the maximal gauge setting. To derive energy estimates for the reduced system at time t, we show that all the terms in the flux integrals on the boundary can be either directly controlled by the boundary conditions or they lead to an integral on the two-dimensional section at time t of the boundary. Exploiting again the maximal gauge condition on the boundary, this contribution to the flux integrals can then be absorbed by a careful trace inequality in the interior energy.
The time-elapsed model for neural assemblies is a nonlinear age-structured equation where the renewal term describes the network activity and influences the discharge rate, possibly with a delay due to the length of connections. We first solve a long standing question, namely that an inhibitory network without delay can promote desynchronization and stabilizes network activity by proving rigorously that the solution converges to a unique steady state. Our approach is based on the observation that a non-expansion property holds. However a non-degeneracy condition is needed and, besides the standard one, we introduce a new condition based on strict nonlinearity. When a delay is included, following previous works for Fokker-Planck models, we prove that the network can generate periodic solutions, both in inhibitory and excitatory networks. To this end, we introduce a new formalism to establish rigorously this property for large delays. Moreover, the fundamental contraction property can extend to other age-structured equations and systems.
