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Two-species system with nonlocal interactions driven by Riesz potentials

Simone Fagioli, Valeria Iorio

Abstract

This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.

Two-species system with nonlocal interactions driven by Riesz potentials

Abstract

This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.

Paper Structure

This paper contains 14 sections, 17 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal transportation theory
  4. Fourier transform and Sobolev Spaces
  5. Main result
  6. Useful inequalities
  7. Energy functional and JKO scheme
  8. Energy functional
  9. JKO-scheme
  10. First flow interchange. The heat flow.
  11. Second flow interchange and L-infinity estimate.
  12. Convergence to weak solutions
  13. Conclusion and perspectives
  14. Useful well-known results

Key Result

Theorem 2.1

Let $T>0$ be fixed, and consider $0<s,r,q< \min \{1, \frac{d}{2} \}$, fixed exponent with Let $(\rho_0, \eta_0) \in X^{s,q}_2\times X^{r,q}_2$. Then, there exists a weak solution $(\rho, \eta)$ to eq:main_syst_1 in the sense of Definition def:weak_sol. Moreover, the energy dissipation inequality is satisfied for all $t\in[0,T]$. Finally, if $(\rho_0, \eta_0) \in L^p(\mathbb{R}^d)\times L^p(\math

Theorems & Definitions (31)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2: Weak solution to \ref{['eq:main_syst_1']}
  • Theorem 2.1
  • Remark 2.2: Choice of $q$
  • Remark 2.3: Case with more than two species
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • ...and 21 more