Analysis and Numerical Approximations of Hydrodynamical Models of Biological Movements

1. Modeling of chemotaxis 1.1 The Patlak-Keller-Segel model 1.2 Hyperbolic models 1.2.1 The Cattaneo-Hillen model 1.2.2 The Gamba-Preziosi model 2. A semilinear hyperbolic-parabolic model of chemotaxis 2.1 Partially dissipative hyperbolic systems 2.1.1 The multidimensional Green function 2.2 Local existence of smooth solutions 2.3 Continuation principle 2.4 Global existence and asymptotic behavior of smooth solutions 2.4.1 Decay estimates for the chemoattractant 2.4.2 Decay estimates for the conservative and dissipative variables 2.5 Global existence and asymptotic behavior of perturbations of constant stationary states 2.5.1 Decay estimates for the conservative and dissipative variables 2.6 Comparison with the Patlak-Keller-Segel model 2.6.1 Asymptotic behavior of the Patlak-Keller-Segel model solutions 2.6.2 Decay estimate of the di↵erence of solutions

[1]  Luigi Preziosi,et al.  Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications , 2009, Journal of mathematical biology.

[2]  F. Bouchut.  Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws , 1999 .

[3]  François Bouchut,et al.  Entropy satisfying flux vector splittings and kinetic BGK models , 2003, Numerische Mathematik.

[4]  A. Coniglio,et al.  Percolation and Burgers' dynamics in a model of capillary formation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  S M Gray,et al.  Estimating a treatment effect from multidimensional longitudinal data. , 1998, Biometrics.

[6]  D. Lauffenburger,et al.  Effects of leukocyte random motility and chemotaxis in tissue inflammatory response. , 1979, Journal of theoretical biology.

[7]  Daniel W. Stroock,et al.  Some stochastic processes which arise from a model of the motion of a bacterium , 1974 .

[8]  R. Natalini.  Convergence to equilibrium for the relaxation approximations of conservation laws , 1996 .

[9]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[10]  Alan D. Lopez,et al.  Global mortality, disability, and the contribution of risk factors: Global Burden of Disease Study , 1997, The Lancet.

[11]  R. Natalini,et al.  Asymptotic high-order schemes for integro-differential problems arising in markets with jumps , 2006 .

[12]  W. Alt.  Biased random walk models for chemotaxis and related diffusion approximations , 1980, Journal of mathematical biology.

[13]  H. Othmer,et al.  Mathematical modeling of tumor-induced angiogenesis , 2004, Journal of mathematical biology.

[14]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes , 2000, SIAM J. Appl. Math..

[15]  C. A. Crispim,et al.  Cyanobacteria and Biodeterioration of Cultural Heritage: A Review , 2003, Microbial Ecology.

[16]  J. Peeters,et al.  A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton , 1988 .

[17]  Thomas Hillen,et al.  Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding , 2001, Adv. Appl. Math..

[18]  L. Segel,et al.  Traveling bands of chemotactic bacteria: a theoretical analysis. , 1971, Journal of theoretical biology.

[19]  M. Rigaud,et al.  Nitric oxide induces cultured cortical neuron apoptosis , 1996, Neuroscience Letters.

[20]  Shuichi Kawashima,et al.  Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation , 1985 .

[21]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[22]  Graeme J. Pettet,et al.  Chemotactic Cellular Migration: Smooth and Discontinuous Travelling Wave Solutions , 2003, SIAM J. Appl. Math..

[23]  Yong Jung Kim.  A MATHEMATICAL INTRODUCTION TO FLUID MECHANICS , 2008 .

[24]  R. H. Adrian,et al.  Action potentials reconstructed in normal and myotonic muscle fibres. , 1976, The Journal of physiology.

[25]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[26]  L. Preziosi,et al.  Mechanics and Chemotaxis in the Morphogenesis of Vascular Networks , 2006, Bulletin of mathematical biology.

[27]  Thomas C. Sideris,et al.  Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping , 2003 .

[28]  Ching-Hong Yang,et al.  Influence of extracellular polymeric substances on Pseudomonas aeruginosa transport and deposition profiles in porous media. , 2007, Environmental science & technology.

[29]  D. Lauffenburger,et al.  Consequences of chemosensory phenomena for leukocyte chemotactic orientation , 1986, Cell Biophysics.

[30]  Helen M Byrne,et al.  A new interpretation of the Keller-Segel model based on multiphase modelling , 2004, Journal of mathematical biology.

[31]  Roberto Natalini,et al.  Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws , 2000, SIAM J. Numer. Anal..

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

[33]  Eugene M. Johnson,et al.  Superoxide dismutase delays neuronal apoptosis: A role for reactive oxygen species in programmed neuronal death , 1995, Neuron.

[34]  D. Dormann,et al.  Chemotactic cell movement during Dictyostelium development and gastrulation. , 2006, Current opinion in genetics & development.

[35]  B. Perthame.  Transport Equations in Biology , 2006 .

[36]  L. Preziosi,et al.  ON THE CLOSURE OF MASS BALANCE MODELS FOR TUMOR GROWTH , 2002 .

[37]  Isaac Klapper,et al.  Mathematical Description of Microbial Biofilms , 2010, SIAM Rev..

[38]  N. Simonian,et al.  Dynamics of polymorphonuclear leukocyte accumulation in acute cerebral infarction and their correlation with brain tissue damage. , 1996, Stroke.

[39]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[40]  Rayleigh.  The Problem of the Random Walk , 1905, Nature.

[41]  M. Chaplain,et al.  Mathematical Modelling of Angiogenesis , 2000, Journal of Neuro-Oncology.

[42]  J. Folkman.  The vascularization of tumors. , 1976, Scientific American.

[43]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[44]  L. Segel,et al.  Incorporation of receptor kinetics into a model for bacterial chemotaxis. , 1976, Journal of theoretical biology.

[45]  R. Natalini.  A Discrete Kinetic Approximation of Entropy Solutions to Multidimensional Scalar Conservation Laws , 1998 .

[46]  DianqingWU.  Signaling mechanisms for regulation of chemotaxis , 2005 .

[47]  J. J. L. Velázquez,et al.  Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions , 2004, SIAM J. Appl. Math..

[48]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations II: Chemotaxis Equations , 2002, SIAM J. Appl. Math..

[49]  P. Krieg,et al.  VEGF mediates angioblast migration during development of the dorsal aorta in Xenopus. , 1998, Development.

[50]  D. Lauffenburger,et al.  Analysis of a lumped model for tissue inflammation dynamics. , 1981, Mathematical biosciences.

[51]  M. Loosdrecht,et al.  Heterotrophic Pioneers Facilitate Phototrophic Biofilm Development , 2007, Microbial Ecology.

[52]  L. Edelstein-Keshet,et al.  Exploring the formation of Alzheimer's disease senile plaques in silico. , 2002, Journal of theoretical biology.

[53]  L. Preziosi,et al.  On the stability of homogeneous solutions to some aggregation models , 2003 .

[54]  Tianyu Zhang,et al.  Phase-Field Models for Biofilms II. 2-D Numerical Simulations of Biofilm-Flow Interaction , 2008 .

[55]  Shuichi Kawashima,et al.  On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws , 1988 .

[56]  C. Schmeiser,et al.  Global existence for chemotaxis with finite sampling radius , 2006 .

[57]  Roberto Natalini,et al.  GLOBAL EXISTENCE OF SMOOTH SOLUTIONS FOR PARTIALLYDISSIPATIVE HYPERBOLIC SYSTEMS WITH A CONVEX ENTROPYB , 2002 .

[58]  Hans G. Othmer,et al.  Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[59]  F. Vilhardt.  Microglia: phagocyte and glia cell. , 2005, The international journal of biochemistry & cell biology.

[60]  L. Preziosi,et al.  On Darcy's law for growing porous media , 2002 .

[61]  Dariusz Wrzosek,et al.  Global attractor for a chemotaxis model with prevention of overcrowding , 2004 .

[62]  M. Block,et al.  Microglia and inflammation-mediated neurodegeneration: Multiple triggers with a common mechanism , 2005, Progress in Neurobiology.

[63]  Howard A. Levine,et al.  A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[64]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[65]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[66]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[67]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[68]  Vincent Calvez,et al.  The parabolic-parabolic Keller-Segel model in R2 , 2008 .

[69]  Thomas Hillen,et al.  Hyperbolic models for chemotaxis in 1-D , 2000 .

[70]  H. Kozono,et al.  Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces , 2009 .

[71]  C. Iadecola,et al.  Cerebral ischemia and inflammation , 2001, Current opinion in neurology.

[72]  Jonathan A. Sherratt,et al.  Chemotaxis and chemokinesis in eukaryotic cells: The Keller-Segel equations as an approximation to a detailed model , 1994 .

[73]  S. Rabouille,et al.  Comparison between two mathematical formulations of the phytoplankton specific growth rate as a function of light and temperature, in two simulation models (ASTER & YOYO) , 2003 .

[74]  A. Karsan,et al.  Signaling pathways induced by vascular endothelial growth factor (review). , 2000, International journal of molecular medicine.

[75]  I. Lapidus,et al.  Model for the chemotactic response of a bacterial population. , 1976, Biophysical journal.

[76]  R. Natalini,et al.  Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis , 2010 .

[77]  G. Feuerstein,et al.  Induced expression of adhesion molecules following focal brain ischemia. , 1995, Journal of neurotrauma.

[78]  R. Natalini,et al.  Convergence of relaxation schemes for conservation laws , 1996 .

[79]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[80]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[81]  Lee A. Segel,et al.  A Theoretical Study of Receptor Mechanisms in Bacterial Chemotaxis , 1977 .

[82]  Takashi Suzuki,et al.  Free Energy and Self-Interacting Particles , 2005 .

[83]  Vicenç Méndez,et al.  Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment , 2002 .

[84]  Yanping Lin,et al.  On the $L^2$-moment closure of transport equations: The Cattaneo approximation , 2004 .

[85]  D A Lauffenburger,et al.  Transient behavior of a chemotaxis system modelling certain types of tissue inflammation , 1987, Journal of mathematical biology.

[86]  R. Kowalczyk,et al.  Preventing blow-up in a chemotaxis model , 2005 .

[87]  K. Todd,et al.  Microglia in cerebral ischemia: molecular actions and interactions. , 2006, Canadian journal of physiology and pharmacology.

[88]  R. Temam.  Une méthode d'approximation de la solution des équations de Navier-Stokes , 1968 .

[89]  Dariusz Wrzosek,et al.  Long-time behaviour of solutions to a chemotaxis model with volume-filling effect , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[90]  Shuichi Kawashima,et al.  Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics , 1984 .

[91]  Alan D. Lopez,et al.  Mortality by cause for eight regions of the world: Global Burden of Disease Study , 1997, The Lancet.

[92]  I. Klapper,et al.  A Multidimensional Multispecies Continuum Model for Heterogeneous Biofilm Development , 2007 .

[93]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[94]  Dirk Horstmann,et al.  Lyapunov functions and $L^{p}$-estimates for a class of reaction-diffusion systems , 2001 .

[95]  Angela Stevens,et al.  The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems , 2000, SIAM J. Appl. Math..