Optimal initial value conditions for the existence of strong solutions of the Boussinesq equations

Uniqueness Criteria and Strong Solutions of the Boussinesq Equations in Completely General Domains

“…The proof is analogous to the proof of [17,Theorem 1.5] with exponents s = 8, q = 4 and s 1 = 8 3 , q 1 = 4. Indeed, since [17, Lemma 3.2] can be replaced by Lemma 3.1 there are no problems occurring in this proof although we consider a general domain instead of a smooth bounded domain in [17,Theorem 1.5].…”

“…Up to now it is not known if weak solutions (u, θ) of the three-dimensional Boussinesq equations are uniquely determined and regular. In [17] it is proved that uniqueness and regularity holds if additionally Serrin's condition u ∈ L s (0, T ; L q (Ω)) holds where 1 < s, q < ∞ with 2 s + 3 q = 1. In general domains, which may have several exits to infinity or may have edges and corners, only the L 2 -approach to the Stokes operator is available.…”

“…The crucial point in the definition above is the fact that we have required no additional integrability condition for θ. It follows from [17,Theorem 1.6] that strong solutions of (1) are smooth if the data are smooth. Further, (see Theorem 4.1 below) strong solutions are uniquely determined.…”

“…For existence of strong solutions of the Boussinesq equations if Ω ⊆ R 3 is a smooth bounded domain we refer to [17,Theorems 1.3,1.4]. From Theorem 1.3 (i) with f 1 := f, f 2 := 0, g := 0 and initial values u 0 and θ 0 := 0 we obtain the following result, c.f.…”

“…From [17,Theorem 1.5] it follows that weak solutions (u, θ) and (v, Θ) of the Boussinesq equations (1) coincide if additionally u, v ∈ L 8 (0, T ; L 4 (Ω)). In the following corollary we will show that the assumption v ∈ L 8 (0, T ; L 4 (Ω)) can be replaced by the weaker assumption that (v, Θ) satisfies the strong energy inequality (9).…”