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Nonlinear biphasic mixture model: existence and uniqueness results

M. Alam, A. Muntean, G. P. Raja Sekhar

TL;DR

This paper derived a weak formulation and then formulated the equivalent fixed-point problem and used the Galerkin method and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

Abstract

This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumor. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study, we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and nonlinear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed point theorems to prove the existence and uniqueness results.

Nonlinear biphasic mixture model: existence and uniqueness results

TL;DR

This paper derived a weak formulation and then formulated the equivalent fixed-point problem and used the Galerkin method and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

Abstract

This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumor. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study, we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and nonlinear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed point theorems to prove the existence and uniqueness results.
Paper Structure (20 sections, 26 theorems, 134 equations, 1 figure, 1 table)

This paper contains 20 sections, 26 theorems, 134 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Biphasic Mixture Theory
  3. Main modeling assumptions
  4. Boundary conditions
  5. Non-dimensional equations
  6. Well-posedness of the auxiliary sub-problems
  7. Concept of weak formulation
  8. Decoupled problem corresponding to \ref{['WFE1']}-\ref{['WFE3']}, Case (a): $\mathbf{K}(\hbox{\boldmath $\varsigma$})=\mathbf{K}(\mathbf{U}_s)$
  9. Decoupled problem corresponding to \ref{['WFE1']}-\ref{['WFE3']}, Case (b): $\mathbf{K}(\nabla\cdot\hbox{\boldmath $\varsigma$})=\mathbf{K}(\nabla\cdot\mathbf{U}_s)$
  10. Case (a): Existence and uniqueness results for $(Q_{w_1}(\hbox{\boldmath $\varsigma$}))$
  11. Case (a): Existence and uniqueness results for $(Q_{w_2}(\mathbf{U}_s))$
  12. Case (b): Existence and uniqueness of a solution to $(Q_{w_1}(\nabla\cdot\hbox{\boldmath $\varsigma$}))$
  13. Case (b): Existence and uniqueness of a solution to $(Q_{w_2}(\nabla\cdot\mathbf{U}_s))$
  14. Case (a): Reduction to a fixed-point problem for $(Q_{w_1}(\mathbf{U}_s))-(Q_{w_2}(\mathbf{U}_s))$
  15. Analysis of the fixed point problem
  16. ...and 5 more sections

Key Result

Lemma 1

(Equivalence of weak formulations) Suppose that parameters and data satisfy assumptions (A) and (B). Then any solution (in the sense of distributions) $(\mathbf{V}_f, \mathbf{U}_s,P)\in H^1(\Omega)^d\times H^1_{0}(\Omega)^d\times L^2(\Omega)$ of the coupled problem (Eq15a)-(Eq18ab) is also a solut

Figures (1)

  • Figure 1: Geometry of the problem

Theorems & Definitions (30)

  • Lemma 1
  • Lemma 3.1
  • Remark 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 20 more