Large-Time Behavior of Solutions to Darcy–Boussinesq Equations with Non-Vanishing Scalar Acceleration Coefficient
We study the large-time behavior of solutions to Darcy–Boussinesq equations with a non-vanishing scalar acceleration coefficient, which model buoyancy-driven flows in porous media with spatially varying gravity. First, we show that the system admits steady-state solutions of the form <inline-form...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/10/1570 |
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| Summary: | We study the large-time behavior of solutions to Darcy–Boussinesq equations with a non-vanishing scalar acceleration coefficient, which model buoyancy-driven flows in porous media with spatially varying gravity. First, we show that the system admits steady-state solutions of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi mathvariant="bold">u</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>,</mo><msub><mi>ρ</mi><mi>s</mi></msub><mo>,</mo><msub><mi>p</mi><mi>s</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>s</mi></msub></semantics></math></inline-formula> is characterised by the hydrostatic balance <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><msub><mi>p</mi><mi>s</mi></msub><mo>=</mo><mo>−</mo><msub><mi>ρ</mi><mi>s</mi></msub><mo>∇</mo><mi mathvariant="normal">Ψ</mi></mrow></semantics></math></inline-formula>. Second, we prove that the steady-state solution satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><msub><mi>ρ</mi><mi>s</mi></msub><mo>=</mo><mi>δ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∇</mo><mi mathvariant="normal">Ψ</mi></mrow></semantics></math></inline-formula> is linearly stable provided that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo><</mo><msub><mi>δ</mi><mn>0</mn></msub><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, while the system exhibits Rayleigh–Taylor instability if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Ψ</mi><mo>=</mo><mi>g</mi><mi>y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>s</mi></msub><mo>=</mo><msub><mi>δ</mi><mn>0</mn></msub><mi>g</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mn>0</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Finally, despite the inherent Rayleigh–Taylor instability that may trigger exponential growth in time, we prove that for any sufficiently regular initial data, the solutions of the system asymptotically converge towards the vicinity of a steady-state solution, where the velocity field is zero, and the new state is determined by hydrostatic balance. This work advances porous media modeling for geophysical and engineering applications, emphasizing the critical interplay of gravity, inertia, and boundary conditions. |
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| ISSN: | 2227-7390 |