Alireza Khatib, Abbas Moameni, Somayeh Mousavinasr
Abstract:
We study the n-dimensional stationary Navier-Stokes equations with a damping
term by developing a new general minimax principle. This principle is sufficiently
broad to be applied in various contexts, and here it is used to establish the existence
of weak solutions for both linear and nonlinear damping, without restrictions on the
damping constant. The damping term, which models physical effects such as porous media
flow, drag, friction, and dissipation, also provides a mathematical advantage by
improving the regularity of solutions compared to the classical Navier-Stokes system.
Our results cover the cases of positive and negative damping constants and
yield existence theorems under different ranges of \(p\) and spatial dimensions.
In particular, we prove solvability even in borderline situations,
such as when \(\mu = -\lambda_1\), where coercivity is lost and traditional minimax
arguments typically fail. The general minimax framework we introduce is flexible and
can be adapted to other nonlinear PDEs, especially when symmetry or structural properties
are involved.
Submitted September 15, 2025. Published January 6, 2026.
Math Subject Classifications: 35Q30, 35A15.
Key Words: Navier-Stokes equation; variational method; damping term.
DOI: 10.58997/ejde.2026.01
Show me the PDF file (363 KB), TEX file for this article.
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Alireza Khatib Universidade Federal do Amazonas Manaus, AM, Brazil email: alireza@ufam.edu.br |
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Abbas Moameni School of Mathematics and Statistics Carleton University Ottawa, ON, Canada email: momeni@math.carleton.ca |
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Somayeh Mousavinasr Universidade Federal do Amazonas Manaus, AM, Brazil email: somayeh@ufam.edu.br |
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