arXiv subjects

S. Sonner

Publications and source records attributed to S. Sonner.

3 recordsLinked to original sources

arXiv2026

Mean-square Stability and Bifurcations for Dissipative SDEs

We investigate the dynamics of dissipative systems with stochastic forcing and focus in particular on mean-square stability. First we show, under a natural condition on the drift and diffusion, that the stochastic system is mean-square dissipative. Next we examine the linearised system and state conditions ensuring that perturbations of a linear system with affine noise are bounded. We then relate the mean-square dynamics of the nonlinear and linearised systems. The approach gives a straightforward deterministic method to examine the effects of stochastic forcing on the stability of equilibria of deterministic systems and to obtain bifurcation diagrams that can be included into standard numerical continuation packages. The technique is illustrated numerically on some standard and non-standard examples.

math.PR↗

arXiv2024

Robust time-discretisation and linearisation schemes for singular and degenerate evolution systems modelling biofilm growth

We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in the system is parabolic and exhibits degenerate and singular diffusion, while the second is either uniformly parabolic or an ordinary differential equation. First, we introduce a semi-implicit time discretisation that has the benefit of decoupling the equations. We prove the positivity, boundedness, and convergence of the time-discrete solutions to the time-continuous solution. Then, we introduce an iterative linearisation scheme to solve the resulting nonlinear time-discrete problems. Under weak assumptions on the time-step size, we prove that the scheme converges irrespective of the space discretisation and mesh. Moreover, if the problem is non-degenerate, the convergence becomes faster as the time-step size decreases. Finally, employing the finite element method for the spatial discretisation, we study the behaviour of the scheme, and compare its performance to other commonly used schemes. These tests confirm that the proposed scheme is robust and fast.

math.NA↗

arXiv2015

Hardy Type Inequalities for $Δ_λ$-Laplacians

We derive Hardy type inequalities for a large class of sub-elliptic operators that belong to the class of $Δ_λ$-Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin type operators $$ Δ_{x}+ |x|^{2α}Δ_{y},\qquad\ (x,y)\in\mathbb{R}^{N_1}\times\mathbb{R}^{N_2},\ α\geq 0, $$ which were proved to be sharp.

math.AP↗