Limiting Theories in Material Science: Mathematical derivation and Analysis

• Applicant: Professor Dr. Peter Bella
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 313878761

• Final Report Year: 2024

Final Report Abstract

The project focused on three areas: pattern formation in wrinkled thin elastic sheets, regularity theory for solutions of elliptic and parabolic equations with degenerate coefficients and their implications, and singular limits in models for motion of compressible viscous fluids. Following previous work on wrinkling of circular sheet on spherical elastic substrate, whose wrinkling is caused by substrate induced curvature, we analyzed the model for the case of a general substrate (including a liquid drop model considered in experiments) and identified the energy scaling law for the leading order energy as well as next-order excess energy. In a different project, we extended the analysis of the optimal prefactor for the excess energy in a problem modeling transition from flat to wrinkled state via Γ-convergence. Starting point of the regularity results is sharp extension of classical work of Trudinger on local boundedness and validity of Harnack inequality for weak solutions of scalar elliptic equations with degenerate coefficients. We have also analyzed parabolic equations, and implications to quenched invariance principle for the random conductance model. Another area it applies to is regularity theory for critical points of double-phase variational integrals, as well as regularity for solutions of the p-Laplacian. In the third area we considered motion of compressible viscous fluids. In a series of papers we considered corresponding Navier-Stokes equations in smooth domains perforated by balls of radius εα and average distance ε. For these we studied the limiting behavior of global weak solutions as ε 0 and obtained different models describing the effective limiting problem depending on the parameter α. We also considered a full Navier-Stokes-Fourier model including temperature of the fluid, and studied two different limits while rescaling both the pressure and the forcing term. For one rescaling we obtained in the limit a well-established the Oberbeck-Boussinesq approximation but with a surprising nonlocal boundary conditions, related to the conservation of the total energy, while other rescaling gives in the limit “pancake model” proposed as a model for atmospheric flows by Majda.

Publications

• Metric-Induced Wrinkling of a Thin Elastic Sheet. Journal of Nonlinear Science, 24(6), 1147-1176.
  Bella, Peter & Kohn, Robert V.
  
• Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors. SIAM Journal on Mathematical Analysis, 49(6), 4658-4703.
  Bella, Peter; Fehrman, Benjamin; Fischer, Julian & Otto, Felix
  
• Wrinkling of a thin circular sheet bonded to a spherical substrate. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 375(2093), 20160157.
  Bella, Peter & Kohn, Robert V.
  
• A Liouville theorem for elliptic systems with degenerate ergodic coefficients. The Annals of Applied Probability, 28(3).
  Bella, Peter; Fehrman, Benjamin & Otto, Felix
  
• A Liouville theorem for stationary and ergodic ensembles of parabolic systems. Probability Theory and Related Fields, 173(3-4), 759-812.
  Bella, Peter; Chiarini, Alberto & Fehrman, Benjamin
  
• Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 13(1), 155-176.
  Bella, Peter & Giunti, Arianna
  
• Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations. Communications on Pure and Applied Mathematics, 74(3), 453-477.
  Bella, Peter & Schäffner, Mathias
  
• On the First Critical Field in the Three Dimensional Ginzburg–Landau Model of Superconductivity. Communications in Mathematical Physics, 367(1), 317-349.
  Román, Carlos
  
• Effective multipoles in random media. Communications in Partial Differential Equations, 45(6), 561-640.
  Bella, Peter; Giunti, Arianna & Otto, Felix
  
• Generalized Multiscale Young Measures. SIAM Journal on Mathematical Analysis, 52(4), 3252-3300.
  Arroyo-Rabasa, Adolfo & Diermeier, Johannes
  
• On the regularity of minimizers for scalar integral functionals with (p,q)-growth. Analysis & PDE, 13(7), 2241-2257.
  Bella, Peter & Schäffner, Mathias
  
• Quenched invariance principle for random walks among random degenerate conductances. The Annals of Probability, 48(1).
  Bella, Peter & Schäffner, Mathias
  
• Non-uniformly parabolic equations and applications to the random conductance model. Probability Theory and Related Fields, 182(1-2), 353-397.
  Bella, Peter & Schäffner, Mathias
  
• Homogenization and Low Mach Number Limit of Compressible Navier-Stokes Equations in Critically Perforated Domains. Journal of Mathematical Fluid Mechanics, 24(3).
  Bella, Peter & Oschmann, Florian
  
• Lipschitz bounds for integral functionals with (p,q)-growth conditions. Advances in Calculus of Variations, 17(2), 373-390.
  Bella, Peter & Schäffner, Mathias
  
• Inverse of Divergence and Homogenization of Compressible Navier–Stokes Equations in Randomly Perforated Domains. Archive for Rational Mechanics and Analysis, 247(2).
  Bella, Peter & Oschmann, Florian
  
• Local boundedness for $ p $-Laplacian with degenerate coefficients. Mathematics in Engineering, 5(5), 1-20.
  Bella, Peter & Schäffner, Mathias
  
• On the Incompressible Limit of a Strongly Stratified Heat Conducting Fluid. Journal of Mathematical Fluid Mechanics, 25(3).
  Basarić, Danica; Bella, Peter; Feireisl, Eduard; Oschmann, Florian & Titi, Edriss S.
  
• Rigorous Derivation of the Oberbeck–Boussinesq Approximation Revealing Unexpected Term. Communications in Mathematical Physics, 403(3), 1245-1273.
  Bella, Peter; Feireisl, Eduard & Oschmann, Florian
  
• Γ–convergence for nearly incompressible fluids. Journal of Mathematical Physics, 64(9).
  Bella, Peter; Feireisl, Eduard & Oschmann, Florian
  
• $$\Gamma $$-Convergence for Plane to Wrinkles Transition Problem. Journal of Nonlinear Science, 35(1).
  Bella, Peter & Marziani, Roberta
  
• Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization. Stochastics and Partial Differential Equations: Analysis and Computations, 12(4), 2246-2288.
  Bella, Peter & Kniely, Michael