Variational Convergence and Stochastic Homogenization of Nonlinear Reaction-Diffusion Problems

Description

A substantial number of problems in physics, chemical physics, and biology, are modeled through reaction-diffusion equations to describe temperature distribution or chemical substance concentration. For problems arising from ecology, sociology, or population dynamics, they describe the density of some populations or species. In this book the state variable is a concentration, or a density according to the cases. The reaction function may be complex and include time delays terms that model various situations involving maturation periods, resource regeneration times, or incubation periods. The dynamics may occur in heterogeneous media and may depend upon a small or large parameter, as well as the reaction term. From a purely formal perspective, these parameters are indexed by n. Therefore, reaction-diffusion equations give rise to sequences of Cauchy problems.The first part of the book is devoted to the convergence of these sequences in a sense made precise in the book. The second part is dedicated to the specific case when the reaction-diffusion problems depend on a small parameter ∊ₙ intended to tend towards 0. This parameter accounts for the size of small spatial and randomly distributed heterogeneities. The convergence results obtained in the first part, with additionally some probabilistic tools, are applied to this specific situation. The limit problems are illustrated through biological invasion, food-limited or prey-predator models where the interplay between environment heterogeneities in the individual evolution of propagation species plays an essential role. They provide a description in terms of deterministic and homogeneous reaction-diffusion equations, for which numerical schemes are possible. Contents Preface 1. Introduction Part 1 Sequences of reaction-diffusion problems: Convergence 2. Variational convergence of nonlinear reaction-diffusion equations 2.1 Existence and uniqueness for reaction-diffusion Cauchy problems in Hilbert spaces 2.1.1 Local existence and uniqueness 2.1.2 Global existence and uniqueness 2.2 Existence and uniqueness of bounded solution of reaction-diffusion problems associated with convex functionals of the calculus of variations and CP-structured reaction functionals 2.2.1 The class of diffusion terms associated with convex integral functionals of the calculus of variations 2.2.2 The class of CP-structured reaction functionals 2.2.3 Examples of CP-structured reaction functionals 2.2.4 The comparison principle 2.2.5 Existence and uniqueness of bounded solutions 2.2.6 Estimate of the L2(Ω)-norm of the right derivative 2.3 Invasion property 2.4 Variational convergence of reaction-diffusion problems with CP-structured reaction functionals 2.5 Stability of the invasion property 2.6 Variational convergence of reaction-diffusion problems: abstract version 3. Variational convergence of nonlinear distributed time delays reaction-diffusion equations 3.1 The time-delays operator 3.1.1 Integration with respect to vector measures 3.1.2 Time-delays operator associated with vector measures 3.1.3 Examples of time-delays operators 3.2 Reaction-diffusion problems associated with convex functionals of the calculus of variations and DCP-structured reaction functionals 3.2.1 The class of DCP-structured reaction functionals 3.2.2 Some examples of DCP-structured reaction functions coming from ecology and biology models 3.2.3 Existence and uniqueness of bounded nonnegative solution 3.3 Convergence theorems 3.3.1 Stability at the limit 3.3.2 An alternative proof of Theorem 3.3 in the case of a single time delay 3.3.3 Non stability of the reaction functional: convergence withmixing effect between growth rates and time delays 4. Variational convergence of two components nonlinear reaction-diffusion systems 4.1 Two components reaction-diffusion system associated with convex functionals of the calculus of variations and TCCP-structured reaction functionals 4.1.1 The class of TCCP-structured reaction functionals 4.1.2 Examples 4.1.3 Existence and uniqueness of a bounded solution 4.2 Convergence theorem of two components reaction-diffusion systems 4.3 Convergence theorem for problems coupling r.d.e. and n.d.r.e. 4.4 Proofs of Propositions 4.1{4.5 4.4.1 Proof of Proposition 4.1 4.4.2 Proof of Proposition 4.2 4.4.3 Proof of Proposition 4.3 4.4.4 Proof of Proposition 4.4 4.4.5 Proof of Proposition 4.5 5. Variational convergence of integrodi erential reaction-diffusion equations 5.1 The general analysis framework 5.1.1 Structure of the first member of (P) 5.1.2 Structure of the reaction functional 5.2 Existence of a local solution 5.2.1 The regularized problem (Pλ) 5.2.2 Convergence of (Pλ) to (P): existence of a local solution of (P) 5.3 Existence of solutions in C([0, T], X) 5.3.1 Existence of a global solution in C([0, T], X): translation-induction method 5.3.2 Existence and uniqueness when Ψ is a quadratic functional 5.3.3 Existence of a right derivative of the solutions at each t ϵ [0, T[ 5.4 Convergence under Mosco×Γ-convergence 5.4.1 The abstract case 5.4.2 The case X = L2(Ω) 6. Variational convergence of a class of functionals indexed by Young measures 6.1 The main continuity result 6.1.1 The lower bound 6.1.2 The upper bound 6.1.3 Proof of Theorem 6.1 6.1.4 Proof of Lemma 6.3 6.2 The case of integral functionals Part 2 Sequences of reaction-diffusion problems: Stochastic homogenization 7. Stochastic homogenization of nonlinear reaction-diffusion equations 7.1 Probabilistic setting 7.1.1 The random diffusion part 7.1.2 The random reaction part 7.2 General homogenization theorems 7.3 Examples of stochastic homogenization of a diffusive Fisher food-limited population model with Allee effect 7.3.1 Random checkerboard-like environment 7.3.2 Environment whose heterogeneities are independently randomly distributed with a frequency λ 7.4 Stochastic homogenization of a reaction-diffusion problem stemming from a hydrogeological model 8. Stochastic homogenization of nonlinear distributed time delays reaction-diffusion equations 8.1 The random diffusion part 8.2 The random reaction part 8.2.1 First structure 8.2.2 Second structure 8.3 Almost sure convergence to the homogenized reaction-diffusion problem 8.4 Application to some examples 8.4.1 Homogenization of vector disease models 8.4.2 Homogenization of delays logistic equations with immigration 8.5 A short digression around percolation for the disease model 8.5.1 Percolation in the "ϵ-random checkerboard-like environment 8.5.2 Percolation in the Poisson point process environment 9. Stochastic homogenization of two components nonlinear reaction-diffusion systems 9.1 The random diffusion parts 9.2 The random reaction parts 9.3 Almost sure convergence to the homogenized system 9.4 The case of a coupling between a random r.d.e. and a random n.d.r.e. 9.5 Application to stochastic homogenization of a prey-predator random model with saturation effect 10. Stochastic homogenization of integrodi erential reaction-diffusion equations 10.1 Stochastic homogenization of a random problem modeled from a Fick's law with delay 10.2 Stochastic homogenization of nonlinear integrodifferential reaction-diffusion equations in one dimension space in the setting of a Poisson point process 11. Stochastic homogenization of non diffusive reaction equations and memory effect 11.1 A general result of homogenization 11.2 Stochastic homogenization of non di usive reaction differential equations: emergence of memory effects Appendix A Grönwall type inequalities Appendix B Basic notions on variational convergences B.1 Γ-convergence B.2 Mosco-convergence B.3 Γ-convergence versus Mosco-convergence B.4 Graph-convergence Appendix C Ergodic theory of subadditive processes C.1 Additive processes C.2 Subadditive processes Appendix D Large deviations principle Appendix E Measure theory E.1 Vector measures E.2 Young measures Appendix F Inf-convolution and parallel sum Bibliography Notation Index