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Long time behaviour of solutions of abstract inequalities. Applications to thermohydraulic and magnetohydrodynamic equations. (English) Zbl 0549.35102
We study some scalar inequalities of parabolic type and we give the leading term of an asymptotic expansion as \(t\to\infty \) for solutions of thermo-hydraulic equations without external excitation. (A phenomenon of resonance is pointed out). We also treat M. H. D. equations, Navier-Stokes equations on a Riemann manifold and scalar inequalities of the type (n(t,x)\(\in {\mathbb{C}}\), \(\Omega\) bounded set in \({\mathbb{R}}^ n):\) \[ |\partial u/\partial t-(\partial /\partial x_ i)(a_{ij}(x,t))\partial u/\partial x_ j|_{L^ 2(\Omega)}\leq n(t)|\nabla u|_{L^ 2(\Omega)^ n} \] where \(a_{ij}(.,t)\) goes to \(a^{\infty}_{ij}(.)\) and n(t) goes to zero in a certain sense when t goes to infinity.
We start with some abstract results on differential inequalities of type: \[ (*)\quad\| (d\Phi /dt)+\nu A\Phi\|_ H\leq n\|\Phi \|_{D(A^{{1\over2}})}\quad (\nu >0), \] where \(\{\) A(t)\(\}\) is a family of self-adjoint unbounded operators on a Hilbert space H. Then these results are applied to the equations and inequalities mentioned previously. The main result for (*) is that the behaviour of \(\Phi\) (t) is characterized by an eigenvalue \(\Lambda^{\infty}\) of the operator \(A^{\infty}=\lim_{t\to +\infty}A(t).\)

MSC:
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35R45 Partial differential inequalities and systems of partial differential inequalities
76W05 Magnetohydrodynamics and electrohydrodynamics
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