Correction to: On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces

We make the following corrections:1. We remove the endpoint results concerning the propagation of the initial regularities (θ2 by the two dimensional viscous Boussinesq flow in [1, Theorem 1.2], which correspond to the two endpoints (2, 0) resp.(1, 2) in [1, Figure 1].See the grey points in Fig. 1 below for the corrected admissible regularity exponent set {(s θ , s uWe remove the two corresponding estimates [1, (1.18) and (1.20)] and their proofs in [1, Subsection 2.3.2 and Subsection 2.3.3] for the two endpoint cases.2. We correct the proof of the uniqueness result in [1, Theorem 1.2] under the initial data assumption, which corresponds to the endpoint (1, 0) in Fig. 1.We remove the second paragraph concerning the optimality of the uniqueness result in [1, Remark 1.3].We correct ν ∈ (s -1, 1) ⊂ (-1, 1) into ν ∈ (max{0, s -1}, 1) in [1, (2.34)].The proofs of the results for the three endpoint regularity cases (s θ , s u ) = (2, 0), or (1, 2), or (1, 0) stated in [1, Theorem 1.2] were wrong due to the failure of the embedding L 1 (R 2 ) → H -1 (R 2 ).We sketch the corrected proofs below, using the same notations as well as the numbering of equations as in [1].