The stochastic p-Laplace equation beyond the classical framework: Well-posedness, memory effects and approximations

• Applicant: Professorin Dr. Aleksandra Zimmermann
• Subject Area: Mathematics
• Term: from 2021 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 490860677

• Final Report Year: 2025

Final Report Abstract

We consider stochastic p-Laplace evolution equations beyond the classical theory of monotone operators. The research project consists of three parts: stochastic p-Laplace equations with memory effects, theoretical aspects of the stochastic p-Laplace equation with convection, discretisation and approximations. The first two parts of the research project have a theoretical focus on the well-posedness of stochastic p-Laplace equations with convection and memory effects. In the third part, full discretisation schemes, semi-implicit with respect to the time variable and of finite-volume type with respect to the space variable, are proposed for stochastic p-Laplace evolution equations and the convergence of these schemes is investigated. The research project is implemented within the framework of a German-French cooperation of outstanding female mathematicians.

Publications

• The stochastic p -Laplace equation on ℝ d. Stochastic Analysis and Applications, 41(5), 892-917.
  Schmitz, Kerstin & Zimmermann, Aleksandra
  
• Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise. ESAIM: Mathematical Modelling and Numerical Analysis, 57(2), 745-783.
  Bauzet, Caroline; Nabet, Flore; Schmitz, Kerstin & Zimmermann, Aleksandra
  
• Finite Volume Approximations for Non-linear Parabolic Problems with Stochastic Forcing. Springer Proceedings in Mathematics & Statistics, 157-166. Springer Nature Switzerland.
  Bauzet, Caroline; Nabet, Flore; Schmitz, Kerstin & Zimmermann, Aleksandra
  
• Stochastic PDEs, finite-volume schemes and applications in mechanics. Mathematics [math]. Habilitation thesis. Aix-Marseille Université, 2023
  C. Bauzet
  
• Diffusion equations with and without memory and stochastic perturbation: theory and numerical approximation. PhD thesis, Universit¨t Duisburg-Essen
  K. Schmitz
  
• Entropy solutions for time-fractional porous medium type equations. Differential and Integral Equations, 37(5/6).
  Schmitz, Kerstin & Wittbold, Petra
  
• A nonlinear stochastic diffusionconvection equation with reflection
  N. Sapountzoglou, Y. Tahraoui, G. Vallet & A. Zimmermann
  
• On a finite-volume approximation of a diffusion-convection equation with a multiplicative stochastic force. Stochastics and Partial Differential Equations: Analysis and Computations, 13(4), 2039-2084.
  Bauzet, Caroline; Schmitz, Kerstin & Zimmermann, Aleksandra
  
• Stochastic pseudomonotone parabolic obstacle problem: well-posedness & Lewy-Stampacchia’s inequalities. Stochastics and Partial Differential Equations: Analysis and Computations.
  Sapountzoglou, Niklas; Tahraoui, Yassine; Vallet, Guy & Zimmermann, Aleksandra
  
• Theoretical analysis of a finite-volume scheme for a stochastic Allen–Cahn problem with constraint. Nonlinear Analysis, 259, 113812.
  Bauzet, Caroline; Sultan, Cédric; Vallet, Guy & Zimmermann, Aleksandra
  
• Well-posedness of stochastic evolution equations with Hölder continuous noise. Electronic Journal of Probability, 30(none).
  Schmitz, Kerstin & Zimmermann, Aleksandra