The Kelvin–Voigt–Brinkman–Forchheimer Equations with Non-Homogeneous Boundary Conditions
Abstract
:1. Introduction
- is the velocity field, ;
- is the pressure, ;
- is the external force field; ;
- is a dimensionless parameter included in the convective term; ;
- is a parameter characterizing the elasticity of the media, ;
- is the viscosity function, ;
- is a function describing hereditary effects (memory), ;
- is a parameter characterizing the permeability of the media, ;
- the symbol ⊗ denotes the tensor product of vectors, that is, for any vectors ;
- the symbol ∇ denotes for the gradient with respect to the space variables , that is, ;
- the differential operators div, and are defined as follows:
2. Preliminaries
2.1. Isomorphisms and Some Related Results
2.2. Local Solvability of Equations with Fréchet Differentiable Operators
- (1)
- The inclusions and hold.
- (2)
- The operator norms and obey the following relation
- (3)
- The mapping is continuously Fréchet differentiable.
- (4)
- The equality holds.
- (5)
- The Fréchet derivative is equal to the zero operator.
2.3. Notation for Scalar Product and Euclidean Norm in and
2.4. Spaces of Time-Independent Functions
- ;
- ;
- the Lebesgue space , and the Sobolev space , .
2.5. Spaces of Time-Dependent Functions
2.6. Helmholtz–Weyl Decomposition and Leray Projection
2.7. Quotient of Sobolev Space by Constants
3. Strong Formulation of Problem
- the vector function and the equivalence class satisfy the system
4. Main Results
- Ω is a bounded domain in space , where or 3, and
- , and
- all inclusions in (13) are valid;
- the vector functions and satisfy the compatibility condition
- (a)
- (b)
5. Auxiliary Propositions
5.1. Boundary Trace and Divergence-Free Lifting
5.2. Two Linear Operators Associated with Kelvin–Voigt–Brinkman–Forchheimer System
6. Proof of Main Theorem
7. Final Comments
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Values of Parameters | Flow Model |
---|---|
, , , , | The incompressible Euler equations for describing unsteady flows of an inviscid fluid [1,2,3,4,5] |
, , , , | The standard Navier–Stokes equations for describing Newtonian fluids motion [6,7,8,9] |
, , , , | The non-stationary Stokes equations for describing the creeping flows of an incompressible viscous fluid [6,7,8,9,10] |
, , , , | The incompressible Navier–Stokes–Voigt equations for describing the flows of Kelvin–Voigt-type viscoelastic fluids [11,12,13,14,15,16] |
, , , , | The Kelvin–Voigt–Brinkman–Forchheimer equations (without an integral memory term) for describing the dynamics of incompressible viscoelastic fluids in a porous medium [17,18,19,20,21,22,23] |
, , , , | The model for describing flows of Jeffreys–Oldroyd-type viscoelastic fluids [24,25,26,27,28] |
, , , , | The Oskolkov integro-differential system for describing the motion of non-Newtonian fluids with memory [29,30,31] |
, , , , | The generalized Navier–Stokes system for describing flows of non-homogeneous incompressible viscous fluids [32,33,34] |
, , , , | The generalized Stokes system for describing creeping flows of non-homogeneous incompressible viscous fluids [35] |
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Baranovskii, E.S.; Artemov, M.A.; Ershkov, S.V.; Yudin, A.V. The Kelvin–Voigt–Brinkman–Forchheimer Equations with Non-Homogeneous Boundary Conditions. Mathematics 2025, 13, 967. https://doi.org/10.3390/math13060967
Baranovskii ES, Artemov MA, Ershkov SV, Yudin AV. The Kelvin–Voigt–Brinkman–Forchheimer Equations with Non-Homogeneous Boundary Conditions. Mathematics. 2025; 13(6):967. https://doi.org/10.3390/math13060967
Chicago/Turabian StyleBaranovskii, Evgenii S., Mikhail A. Artemov, Sergey V. Ershkov, and Alexander V. Yudin. 2025. "The Kelvin–Voigt–Brinkman–Forchheimer Equations with Non-Homogeneous Boundary Conditions" Mathematics 13, no. 6: 967. https://doi.org/10.3390/math13060967
APA StyleBaranovskii, E. S., Artemov, M. A., Ershkov, S. V., & Yudin, A. V. (2025). The Kelvin–Voigt–Brinkman–Forchheimer Equations with Non-Homogeneous Boundary Conditions. Mathematics, 13(6), 967. https://doi.org/10.3390/math13060967