Kinetic-fluid derivation and mathematical analysis of a nonlocal cross-diffusion–fluid system

Abstract In this article we propose a nonlocal cross-diffusion–fluid system describing the dynamics of multiple interacting populations living in a Newtonian fluid. First, we derive our nonlocal cross-diffusion–fluid system from a nonlocal kinetic-fluid model by the micro-macro decomposition method. Second, we prove the existence of weak solutions for the proposed system by applying the nonlinear Galerkin method with a priori estimates and compactness arguments. On the basis of micro-macro decomposition, we propose and develop an asymptotic-preserving numerical scheme. Finally, we discuss the computational results for the proposed system.

[1]  Nicola Bellomo,et al.  Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues , 2015 .

[2]  S. N. Kruzhkov,et al.  Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications , 1969 .

[3]  José A. Carrillo,et al.  An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic Systems for Chemotaxis , 2013, Multiscale Model. Simul..

[4]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[5]  M. Sepúlveda,et al.  Numerical analysis for a three interacting species model with nonlocal and cross diffusion , 2015 .

[6]  Abdelghani Bellouquid,et al.  ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION , 2010 .

[7]  Horst Malchow,et al.  Spatiotemporal Complexity of Plankton and Fish Dynamics , 2002, SIAM Rev..

[8]  Ansgar Jüngel,et al.  Analysis of a Multidimensional Parabolic Population Model with Strong Cross-Diffusion , 2004, SIAM J. Math. Anal..

[9]  Alan Hastings,et al.  Chaos in three species food chains , 1994 .

[10]  D. Burini,et al.  Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles , 2017 .

[11]  A. Hastings,et al.  Chaos in a Three-Species Food Chain , 1991 .

[12]  Shengmao Fu,et al.  Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics , 2009 .

[13]  Esther S. Daus,et al.  Global Existence Analysis of Cross-Diffusion Population Systems for Multiple Species , 2016, 1608.03696.

[14]  M. Sepúlveda,et al.  A NUMERICAL ANALYSIS OF A REACTION–DIFFUSION SYSTEM MODELING THE DYNAMICS OF GROWTH TUMORS , 2010 .

[15]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[16]  Kevin S. McCann,et al.  Bifurcation Structure of a Three-Species Food-Chain Model , 1995 .

[17]  Alexander Lorz,et al.  COUPLED CHEMOTAXIS FLUID MODEL , 2010 .

[18]  Luc Mieussens,et al.  A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit , 2008, SIAM J. Sci. Comput..

[19]  Diego Alexander Garzón-Alvarado,et al.  Computational examples of reaction–convection–diffusion equations solution under the influence of fluid flow: First example , 2012 .

[20]  B. Perthame,et al.  Derivation of hyperbolic models for chemosensitive movement , 2005, Journal of mathematical biology.

[21]  Nicolas Crouseilles,et al.  Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part I: The Case of Heavy-Tailed Equilibrium , 2016, SIAM J. Sci. Comput..

[22]  M. Lachowicz,et al.  Methods of Small Parameter in Mathematical Biology , 2014 .

[23]  J. Deteix,et al.  A Coupled Prediction Scheme for Solving the Navier-Stokes and Convection-Diffusion Equations , 2014, SIAM J. Numer. Anal..

[24]  N. Shigesada,et al.  Biological Invasions: Theory and Practice , 1997 .

[25]  Georges Chamoun,et al.  A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion , 2014, Comput. Math. Appl..

[26]  Anotida Madzvamuse,et al.  Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations , 2015, Journal of mathematical biology.

[27]  Frank Jenko,et al.  How turbulence regulates biodiversity in systems with cyclic competition. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  T. Chung.  Computational Fluid Dynamics: FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES , 2002 .

[29]  Aurel Wintner,et al.  Asymptotic Integrations of Linear Differential Equations , 1955 .

[30]  Ansgar Jüngel,et al.  Entropy Methods for Diffusive Partial Differential Equations , 2016 .

[31]  Joydev Chattopadhyay,et al.  Interactive effects of prey refuge and additional food for predator in a diffusive predator-prey system , 2017 .

[32]  Min Zhao,et al.  Chaos in a three-species food chain model with a Beddington–DeAngelis functional response ☆ , 2009 .

[33]  Nicola Bellomo,et al.  From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid , 2016 .

[34]  Ayman Moussa,et al.  Entropy, Duality, and Cross Diffusion , 2013, SIAM J. Math. Anal..

[35]  Ansgar Jüngel,et al.  Analysis of a parabolic cross-diffusion population model without self-diffusion , 2006 .

[36]  Juan Soler,et al.  Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker–Smale models , 2016, 1611.00743.

[37]  John W. Barrett,et al.  Finite element approximation of a nonlinear cross-diffusion population model , 2004, Numerische Mathematik.

[38]  Mostafa Bendahmane,et al.  Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system , 2018, Mathematical Methods in the Applied Sciences.

[39]  Michael Winkler,et al.  Stabilization in a two-dimensional chemotaxis-Navier–Stokes system , 2014, 1410.5929.

[40]  Canrong Tian,et al.  Instability induced by cross-diffusion in reaction–diffusion systems , 2010 .

[41]  Ricardo Ruiz-Baier,et al.  Analysis of a finite volume method for a cross-diffusion model in population dynamics , 2011 .

[42]  M. Chipot,et al.  Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems , 2001 .

[43]  Long Chen.  FINITE VOLUME METHODS , 2011 .

[44]  A. Jüngel.  Diffusive and nondiffusive population models , 2010 .

[45]  Edmund J. Crampin,et al.  Reaction-Diffusion Models for Biological Pattern Formation , 2001 .

[46]  N. Bellomo,et al.  On the onset of non-linearity for diffusion models of binary mixtures of biological materials by asymptotic analysis , 2006 .

[47]  Luc Mieussens,et al.  An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit , 2009 .

[48]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[49]  Luc Mieussens,et al.  Macroscopic Fluid Models with Localized Kinetic Upscaling Effects , 2006, Multiscale Model. Simul..

[50]  M. C. Lombardo,et al.  A velocity--diffusion method for a Lotka--Volterra system with nonlinear cross and self-diffusion , 2009 .

[51]  Mostafa Bendahmane,et al.  Weak and classical solutions to predator―prey system with cross-diffusion , 2010 .

[52]  J. Schnakenberg,et al.  Simple chemical reaction systems with limit cycle behaviour. , 1979, Journal of theoretical biology.

[53]  Juan Soler,et al.  MULTISCALE BIOLOGICAL TISSUE MODELS AND FLUX-LIMITED CHEMOTAXIS FOR MULTICELLULAR GROWING SYSTEMS , 2010 .