On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case
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| Format: | Preprint |
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2026
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| _version_ | 1866909999106621440 |
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| author | Abbatiello, Anna Bulíček, Miroslav Kaplický, Petr |
| author_facet | Abbatiello, Anna Bulíček, Miroslav Kaplický, Petr |
| contents | We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold.
Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16868 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case Abbatiello, Anna Bulíček, Miroslav Kaplický, Petr Analysis of PDEs 35Q30, 35K61, 76E30, 37L15 We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold. Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime. |
| title | On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case |
| topic | Analysis of PDEs 35Q30, 35K61, 76E30, 37L15 |
| url | https://arxiv.org/abs/2601.16868 |