Large time decay and growth for solutions of a viscous Boussinesq system

By Lorenzo Brandolese, Maria E. Schonbek. 2010

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In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as t→∞ in the sense that the energy and the L-norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from (u0, θ0) with zero-mean for the initial temperature θ0 have a special behavior as |x| or t tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/88886f5996b6ea08a4d83e60437169d3a6822790