Asymptotic stability estimates for some evolution problems with singular convection field
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Asymptotic stability estimates for some evolution problems with singular convection field Fernando Farroni1 Received: 12 June 2020 / Accepted: 22 September 2020 © The Author(s) 2020
Abstract We establish asymptotic stability estimates for solutions to evolution problems with singular convection term. Such quantitative estimates provide a measure with respect to the time variable of the distance between the solution to a parabolic problem from the one of the its elliptic stationary counterpart. Keywords Evolution problems · Asymptotic estimates · Convection term Mathematics Subject Classification 35K45
1 Introduction This paper concerns evolution problems whose model case reads as follows ⎧ ⎪ ⎨ u t − div M(x, t)∇u + A |x|x 2 u = − div F u=0 on ∂ × (0, T ), ⎪ ⎩ in , u(x, 0) = u 0 (x)
in T , (1.1)
Here and in what follows denotes a regular bounded domain of R N with N ≥ 3, A > 0, T ∈ (0, ∞] and T stands for the cylinder × (0, T ). With regard to the structure assumptions of the problem, we assume that M = M(x, t) : × (0, T ) →
The Author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
B 1
Fernando Farroni [email protected] Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli Studi di Napoli Federico II, Complesso Monte S. Angelo, via Cinthia, 80126 Naples, Italy
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F. Farroni
R N ×N is a measurable, symmetric, matrix field satisfying the uniform bounds λ|ξ |2 ≤ M(x, t)ξ, ξ ≤ |ξ |2
(1.2)
for every ξ ∈ R N and for a.e. (x, t) ∈ × (0, T ) where 0 < λ ≤ . For the data of the problem we assume that F ∈ L 2 T , R N
and
u 0 ∈ L 2 ()
(1.3)
The aim of this note is to provide a quantitative estimate related to the long time behavior of the global in time weak solution u = u(x, t) of (1.1) (according to Definition 3.1 below). As an example, we wonder whether the solution u = u(x, t) defined in the whole of ∞ tends toward the one of the stationary problem
−v − div A |x|x 2 v = − div f v=0 on ∂,
in ,
(1.4)
as t → ∞. For the data and for the structure assumptions relative to problem (1.4), we assume f ∈ L 2 (, R N ) If all the assumptions above are fulfilled, an important property for the elliptic problem (1.4) relies on the fact that that if a solution exists then it is automatically unique (see e.g. [16]). Obseve that our problem exhibits an unbounded and singular convection term if 0 ∈ , because of the presence of coefficient E A (x) := A |x|x 2 . We introduce the following functions K (t) := 1 + M(·, t) − I L ∞ () H0 (t) := F(·, t) − f L 2 ()
(1.5) (1.6)
which can be read as measures in time of the distances between the matrix M and the identity I and F and f respectively. We assume that there exists t0 ∈ [0, T )such that (K − 1)2 , H02 ∈ L 1 ([t0 , T ))
(1.7)
Finally, we set K := (K − 1)2 ∇v 2L 2 () + H02 We assume that contains the origin (so that the coefficient E A is singular) and we state our resul
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