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Homogenization results for the generator of multiscale Langevin dynamics in weighted Sobolev spaces

Andrea Zanoni

Abstract

We study the homogenization of the Poisson equation with reaction term and of the eigenvalue problem associated to the generator of multiscale Langevin dynamics. Our analysis extends the theory of two-scale convergence to the case of weighted Sobolev spaces in unbounded domains. We provide convergence results for the solution of the multiscale problems above to their homogenized surrogate. A series of numerical examples corroborate our analysis.

Homogenization results for the generator of multiscale Langevin dynamics in weighted Sobolev spaces

Abstract

We study the homogenization of the Poisson equation with reaction term and of the eigenvalue problem associated to the generator of multiscale Langevin dynamics. Our analysis extends the theory of two-scale convergence to the case of weighted Sobolev spaces in unbounded domains. We provide convergence results for the solution of the multiscale problems above to their homogenized surrogate. A series of numerical examples corroborate our analysis.
Paper Structure (14 sections, 15 theorems, 159 equations, 4 figures)

This paper contains 14 sections, 15 theorems, 159 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Outline.
  4. Problem setting
  5. Preliminary results
  6. Poisson equation with a reaction term
  7. Weak formulation
  8. Two-scale convergence
  9. Homogenization
  10. Eigenvalue problem
  11. Numerical illustration
  12. Poisson problem with a reaction term
  13. Eigenvalue problem
  14. Conclusion

Key Result

Lemma 2.3

Under ass:dissipativity, there exist two constants $C_{\mathrm{low}}, C_{\mathrm{up}} > 0$ independent of $\varepsilon$ such that In particular, the injections $I_{L^2_{\rho^\varepsilon}(\mathbb{R}^d) \hookrightarrow L^2_{\rho^0}(\mathbb{R}^d)}$ and $I_{L^2_{\rho^0}(\mathbb{R}^d) \hookrightarrow L^2_{\rho^\varepsilon}(\mathbb{R}^d)}$ are continuous.

Figures (4)

  • Figure 1: Multiscale and homogenized solution of the Poisson problem with a reaction term setting $\varepsilon = 0.1$.
  • Figure 2: Poisson problem with a reaction term varying $\varepsilon$. Left: distance between the multiscale and homogenized solution. Right: distance between the multiscale solution and its first order expansion.
  • Figure 3: First four eigenvalues and eigenfunctions of the multiscale and homogenized generator setting $\varepsilon = 0.1$.
  • Figure 4: Distance between the first four eigenvalues and eigenfunctions of the multiscale and homogenized generator varying $\varepsilon$.

Theorems & Definitions (32)

  • Remark 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 22 more