Joint research project

Some classes of PDE and systems with applications to mechanics and biology

Project leaders
Roberto Natalini, George Jaiani
Agreement
GEORGIA - SRNSF - Shota Rustaveli National Science Foundation
Call
CNR/SRNSF 2012-2013
Department
Materials and Devices
Thematic area
Physical sciences and technologies of matter
Status of the project
New

Research proposal

The bilateral project is devoted to investigation of some partial differential equations (PDEs) and systems arising in mechanics and biology. Namely, IAC-CNR Team will study hyperbolic models of partial differential equations arising from biological problems.
- Our first goal is to establish some results of global existence and asymptotic behavior for a general class of systems arising to describe the motion of cell aggregates (bacteria, eukaryotic cells) under different kinds of external stimuli (chemical, light, contact, etc...). The considered models arise from Mixture theory along the line proposed by Preziosi and co., and range from classical models of vasculogenesis, to the more recent multiphase models proposed in [CDNR09] to study the growth of biofilms. The technical tools involved will be essentially based on a careful study of the homogeneous problem, i.e.: without the source terms, and of its interaction with the contribution of the source term. Asymptotic behavior will be studied by using a detailed study of the Green function of the linearized problem and Duhamel formulas, as done in the dissipative case in [BHN07, GMNR09]. We also aim to study the pattern formation phenomena for this models, which are observed in some numerical simulations. In particular, we propose to consider more general models, in several dimensions, and with an arbitrary, but finite number of velocities, the aim being to rigorously determine a sharp threshold describing the transition to instability and the arising of non-homogenous pattern formation.
- Also we intend to develop some models of the immunitary response to bacterial infections. A first attempt, using a mixing of classical diffusive chemotaxis models and prey-predator source interactions, is contained in [CN10]. We shall consider suitable hyperbolic generalizations of these models, which are strongly connected with population dynamics and game theory, as in [ABN10]. The global boundedness and asymptotic behavior of solutions will be studied, with a special care in the understanding the phenomena of the appearance of cooperation in the spatial version of classical problems arising in game theory (Prisoners' dilemma or Snowdrift problem).
Meanwhile Georgian team will construct and investigate two-dimensional (2D) hierarchical models for prismatic shells on the bases of thermal elasticity with microtemperatures. The dynamical governing system of the linear theory of elastic materials with inner structure whose particles in addition to the classical displacement and temperature fields possess microtemperature was constructed by Iesan and Quintanilla [1] in 2000. The works of Scalia, Svanadze, and Tracina (see, e.g. [2,3]) are devoted to the investigation of this three-dimensional (3D) model. Each of the hierarchical models which will be constructed for prismatic thermoelastic shells with microtemperatures can be considered as an independent physical model. The researchers of the Georgian team have an experience of construction and investigation of hierarchical models of elastic prismatic and standard shells [4-6], in particular, with the thickness vanishing (so called cusped shells) on the boundary (see [7] and references therein). The outcomes will be: existence and uniqueness theorems for boundary value problems (BVPs) and initial-boundary value problems (IBVPs) within the framework of 2D hierarchical models of the prismatic thermoelastic shells with microtemperatures; proof of the convergence of the sequence of approximate solutions of three space variables restored from the solutions of BVPs and IBVPs corresponding to the constructed 2D hierarchical models to the exact solutions of the original 3D BVPs and IBVPs; in the lower order approximations (hierarchical models) the peculiarities of setting boundary conditions (BCs) depending on the geometry of the sharpening of the cusped edged of the prismatic thermoelastic shells with microtemperatures will be studied. During the cooperation of the Italian and Georgian teams the background of outlooks for further cooperation in the study of the geometrically and physically nonlinear prismatic thermoelastic shells with microtemperatures will be prepared. That will be achieved by the systematic joint discussions of tasks by both the Italian and Georgian teams by electronic means and during mutual visits. Mathematically the above nonlinear problems will lead to the analysis of BVPs and IBVPs of nonlinear PDEs and systems. The cooperation of the Italian and Georgian teams will build also on a previous common experience on the theme of hierarchical models of prismatic shells [CJMPG11].
IAC-CNR Team: [CDNR09] Fabrizio Clarelli, Cristiana Di Russo, Roberto Natalini, Magali Ribot, Mathematical models for biofilms on the surface of monuments, to appear in applied and industrial mathematics in Italy, proceedings of SIMAI Conference 2008, Series on Advances in Mathematics for Applied Sciences - Vol. 82, World Scientific 2009.
[BHN07] S. Bianchini, B. Hanouzet, R. Natalini. Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math. 60, (2007), 1559-1622
[GMNR09] F. Guarguaglini, C. Mascia, R. Natalini, M. Ribot, Global stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model for chemotaxis, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 1, 39-76.
[CN10] F. Clarelli, R. Natalini, A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions. Mathematical Biosciences and Engineering, Volume: 7 Issue: 2 Pages: 277-300, 2010.
[CJMPG11] N. Chinchalaze, G. Jaiani, B. Maistrenko, P. Podio-Guidugli, Concentrated contact interactions in cuspidate prismatic shells. Archive of Applied Mechanics, Volume 81, pp. 1487-505, 2011.
GEORGIAN Team:
• Iesan D. and Quintanilla R. (2000) On a theory of thermoelasticity with microtemperatures. J. of Thermal Stresses: vol. 23, pp. 199-215.
• Svanadze M. (2004) Fundamental Solutions of the Equations of the Theory of Thermoelasticity with Microtemperatures. J. of Thermal Stresses: vol. 27, pp. 151-170.
• Scalia A., Svanadze M., and Tracina R. (2010) Basic theorems in the equilibrium theory of thermoelasticity with microtemperatures. J. of Thermal Stresses: vol. 33, 721-753.
• Gordeziani, D.G. (1974) To the exactness of one variant of the theory of thin shells. Soviet. Math. Dokl.: vol. 215 (4), pp. 751-754.
• Gordeziani, D.G. (1974) On the solvability of some boundary value problems for a variant of the theory of thin shells (Russian). Dokl. Akad. Nauk SSSR: vol. 215 (6), pp. 1289-1292.
• Avalishvili, G., Avalishvili, M., Gordeziani, D., and Miara, B. (2010) Hierarchical modeling of thermoelastic plates with variable thickness. Anal. Appl.: vol. 8 (2), pp. 125-159.
• Jaiani G. Cusped Shell-like Structures, SpringerBriefs in Applied Science and Technology, Springer-Heidelberg-Dordrecht-London-New York, 2011.
 

Research goals

- Global existence and asymptotic behavior for a general class of systems arising to describe the motion of cell aggregates (bacteria, eukaryotic cells) under different kinds of external stimuli (chemical, light, contact, etc...) will be established.
- Construction of differential hierarchical models for prismatic thermoelastic shells with microtemperatures. Proof of the existence and uniqueness theorems of BVPs for the Nth order approximation (hierarchical model). Preparation of a paper for publication. - General models, in several dimensions, and with an arbitrary, but finite number of velocities will be considered.
- Proof of the convergence of the sequence of approximate solutions of three space variables restored from the solutions of BVPs corresponding to the constructed 2D hierarchical models to the exact solutions of the original 3D BVPs. In the lower order approximations (hierarchical models) the peculiarities of setting boundary conditions (BCs) depending on the geometry of the sharpening of the cusped edged of the prismatic thermoelastic shells with microtem¬peratures will be studied. Publication of a paper.
- Some models of the immuni response to bacterial infections will be developed.
- The global boundedness and asymptotic behavior of solutions will be studied, with a special care in the understanding the phenomena of the appearance of cooperation in the spatial version of classical problems arising in game theory (Prisoners' dilemma or Snowdrift problem).
Proof of the convergence of the sequence of approximate solutions of three space variables restored from the solutions of IBVPs corresponding to the constructed 2D hierarchical models to the exact solutions of the original 3D IBVPs. In the lower order approximations (hierarchical models) IBVP-s for cusped prismatic thermoelastic shells with microtem¬peratures will be studied. Publication of a paper.
On the basis of hierarchical models the stress-strain state of prismatic shells, taking into account temperature fields and biofilms arising on the face surfaces of the prismatic shell, will be studied.
- Proof of the existence and uniqueness theorems of IBVPs for the Nth order approximation (hierarchical model). Preparation of a paper for publication.
 

Last update: 28/03/2024