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

Correction to: Z. Angew. Math. Phys. (2022) 73:16 https://doi.org/10.1007/s00033-021-01650-3

We make the following corrections:

• 1. We remove the endpoint results concerning the propagation of the initial regularities $(\theta _0,u_0)\in H^2({\mathbb {R}}^2)\times (L^2({\mathbb {R}}^2))^2\text { resp. }H^1({\mathbb {R}}^2)\times (H^2({\mathbb {R}}^2))^2$ 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 $\{(s_\theta , s_u)\in [1,\infty )\times [0,\infty )\,|\, s_u-1\leqslant s_\theta \leqslant s_u+2\}\backslash \{(2,0), (1,2)\}$.

  We 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 $(\theta _0,u_0)\in H^1({\mathbb {R}}^2)\times (L^2({\mathbb {R}}^2))^2$, 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 $\nu \in (s-1,1)\subset (-1,1)$ into $\nu \in (\text {max}\{0,s-1\},1)$ in [1, (2.34)].

Fig. 1

Admissible regularity exponents

The proofs of the results for the three endpoint regularity cases $(s_\theta , s_u)=(2,0)$, or (1, 2), or (1, 0) stated in [1, Theorem 1.2] were wrong due to the failure of the embedding $L^1({\mathbb {R}}^2)\not \hookrightarrow H^{-1}({\mathbb {R}}^2)$. We sketch the corrected proofs below, using the same notations as well as the numbering of equations as in [1].

• 1. Since we remove the two endpoints (2, 0) and (1, 2) in Fig. 1, that is, the two (technical) inequalities in [1, (1.18) and (1.20)]:

  • $H^2$-Estimate for $\theta$, if $u\in L^\infty _\text {loc}([0,\infty ); (L^2({\mathbb {R}}^2))^2)\cap L^2_\text {loc}([0,\infty ); (H^1({\mathbb {R}}^2))^2)$;

  • $H^2$-Estimate for u, if $\theta \in L^\infty _\text {loc}([0,\infty ); H^1({\mathbb {R}}^2))\cap L^2_\text {loc}([0,\infty ); H^2({\mathbb {R}}^2))$,

  we have to show

  • $H^2$-Estimate for $\theta$, if $u\in L^\infty _\text {loc}([0,\infty ); (H^{0_+}({\mathbb {R}}^2))^2)\cap L^2_\text {loc}([0,\infty ); (H^{1_+}({\mathbb {R}}^2))^2)$;

  • $H^2$-Estimate for u, if $\theta \in L^\infty _\text {loc}([0,\infty ); H^{1_+}({\mathbb {R}}^2))\cap L^2_\text {loc}([0,\infty ); H^{2_+}({\mathbb {R}}^2))$,

  such that the vertical line $\{(2, s_u)\,|\, s_u\in (0,2]\}$ and the horizontal line $\{(s_\theta ,2)\,|\, s_\theta \in (1,2]\}$ are included in the admissible regularity exponent set. More precisely, if $u\in L^\infty _\text {loc}([0,\infty ); (H^{\varepsilon }({\mathbb {R}}^2))^2)\cap L^2_\text {loc}([0,\infty ); (H^{1+\varepsilon }({\mathbb {R}}^2))^2)$ for some $\varepsilon \in (0,1)$, then we have (instead of [1, (1.18)])

  $$\begin{aligned} \begin{aligned}&\Vert \theta \Vert _{L^\infty _T H^2_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^2_x}^2 \leqslant C(\kappa _*,\Vert a\Vert _{C^2}, \kappa ^*) \Vert \theta _0\Vert ^2_{H^2} (1+\Vert \nabla \theta _0\Vert ^2_{L^2}) \\&\quad \times \exp \Bigl (C(\kappa _*,\varepsilon , \Vert a\Vert _{\textrm{Lip}})(\Vert u\Vert _{L^2_T H^{1+\varepsilon }_x}^2+\Vert u\Vert _{L^4_T L^4_x}^4+\Vert \nabla \theta \Vert _{L^4_T L^4_x}^4)\Bigr ) , \end{aligned} \end{aligned}$$

  ((1.18)')

  which follows from the same argument as in [1, Subsection 2.3.2], but with the following inequality (instead of [1, the inequality at the top of Page 16])

  $$\begin{aligned} \begin{aligned} \int \limits _{{\mathbb {R}}^2}|\nabla \Delta \eta \cdot \nabla u\cdot \nabla \eta |\,dx&\leqslant \frac{\kappa ^*}{4}\Vert \nabla \Delta \eta \Vert _{L^2_x}^2 +C(\kappa ^*)\Vert \nabla u\cdot \nabla \eta \Vert _{L^2_x}^2\\&\leqslant \frac{\kappa ^*}{4}\Vert \nabla \Delta \eta \Vert _{L^2_x}^2 +C(\kappa ^*)\Vert \nabla u\Vert _{L^{\frac{2}{1-\varepsilon }}_x}^2\Vert \nabla \eta \Vert _{L^{\frac{2}{\varepsilon }}_x}^2\\&\leqslant \frac{\kappa ^*}{4}\Vert \nabla \Delta \eta \Vert _{L^2_x}^2 +C(\kappa ^*, \varepsilon )\Vert \nabla u\Vert _{H^\varepsilon _x}^2\Vert \nabla \eta \Vert _{H^1_x}^2. \end{aligned} \end{aligned}$$

  Similarly, we have (instead of [1, (1.20)])

  $$\begin{aligned} \begin{aligned}&\Vert u\Vert _{L^\infty _T H^2_x}^2+\Vert \nabla u\Vert _{L^2_T H^2_x}^2 \leqslant (\Vert u\Vert _{L^\infty _T H^1_x}^2+\Vert \nabla u\Vert _{L^2_T H^1_x}^2)\\&\quad +C\Bigl (\Vert \Delta u_0\Vert _{L^2_x}^2+\Vert u\Vert _{L^\infty _T H^1_x\cap L^2_T \dot{H}^2_x}^2 (\Vert u\Vert _{L^\infty _T H^1_x\cap L^2_T \dot{H}^2_x}^2+\Vert \nabla \theta \Vert _{L^2_T H^{1+\varepsilon }_x}^2) \\&\quad +\Vert \Delta \theta \Vert _{L^2_T L^2_x}\Vert \Delta u\Vert _{L^2_T L^2_x}\Bigr ) \times \exp \Bigl (C \bigl (\Vert (u, \nabla \theta )\Vert _{L^4_T L^4_x}^4+\Vert \nabla ^2\theta \Vert _{L^2_T H^{\varepsilon }_x}^2 \bigr )\Bigr ). \end{aligned} \end{aligned}$$

  ((1.20)')

  where the constant C depends on $\mu _*, \varepsilon , \Vert b\Vert _{C^2},\Vert \theta \Vert _{L^\infty _TH^{1+\varepsilon }_x},\Vert \nabla \theta \Vert _{L^2_T H^1_x}$. We remark here that in general we can not show $\nabla \Delta \eta \cdot \nabla u\cdot \nabla \eta \in L^1_\text {loc}L^1_x$ if $(\eta , u)\in L^\infty _\text {loc}(H^2_x\times (L^2_x)^2)\cap L^2_\text {loc}(H^3_x\times (H^1_x)^2)$ because of the failure of the Sobolev embedding $H^1({\mathbb {R}}^2)\not \hookrightarrow L^\infty ({\mathbb {R}}^2)$.

• 2. We now correct the proof of the uniqueness result with initial data $(\theta _0,u_0)\in H^1({\mathbb {R}}^2)\times (L^2({\mathbb {R}}^2))^2$ given in [1, Section 2.2]. We are going to show $H^{1+\delta }\times H^\delta$, $\delta \in (-1,0)$-Estimates (instead of $H^1\times L^2$-Estimates in [1]) for the difference $({\dot{\theta }}, \dot{u})$ of two weak solutions $(\theta _1, u_1)$ and $(\theta _2, u_2)$ satisfying

  $$\begin{aligned}&\theta _1, \theta _2\in C([0,\infty );H^1({\mathbb {R}}^2))\cap L^2_\text {loc}([0,\infty );H^2({\mathbb {R}}^2)),\\&u_1, u_2\in C([0,\infty );(L^2({\mathbb {R}}^2))^2)\cap L^2_\text {loc}([0,\infty );(H^1({\mathbb {R}}^2))^2), \end{aligned}$$

  following the arguments in [1, Subsection 2.3.1]. More precisely, we first observe that by virtue of the estimates in [1, (1.13) and (1.14)],

  $$\begin{aligned} B(t)&:= 1+ \Vert (\nabla u_1, \nabla u_2)\Vert _{L^2_x}^2+\Vert (u_1, u_2, \nabla \eta _1, \nabla \eta _2)\Vert _{H^{\frac{1}{2}}_x}^4 +\Vert (\nabla \eta _1, \nabla \eta _2)\Vert _{H^1_x}^2 \\&\in L^1_\text {loc}([0,\infty )), \end{aligned}$$

  where $\eta =A(\theta ):=\int \limits ^\theta _0 a(\alpha ) d\alpha$ is the function introduced in [1, (2.10)]. Following the arguments in the proof of [1, Lemma 2.2, Subsection 2.3.1], we derive the $H^{\delta +1}\times H^{\delta }$-Estimates for $({\dot{\eta }}, \dot{u})$ which satisfies the equations in [1, (2.19)], in the following three steps:

  • Step 1. We use the commutator estimate in [1, (2.28)] for $\delta \in (-1,0)$ and $j\geqslant -1$

    $$\begin{aligned} \begin{aligned} \Vert [u_1,\Delta _j]\nabla {\dot{\eta }}\Vert _{L^2_x}&\leqslant Cl_j2^{-j\delta } \Vert \nabla u_1\Vert _{L^2_x}\Vert \nabla {\dot{\eta }}\Vert _{H^{\delta }_x},\\ \Vert [\kappa _1,\Delta _j]\Delta {\dot{\eta }}\Vert _{L^2_x}&\leqslant Cl_j2^{-j\delta } \Vert \nabla \kappa _1\Vert _{H^{\frac{1}{2}}_x}\Vert \Delta {\dot{\eta }}\Vert _{H^{\delta -\frac{1}{2}}_x}, \end{aligned} \end{aligned}$$

    with $(l_j)_{j\geqslant -1}\in \ell ^1$, and the product estimates

    $$\begin{aligned} \Vert {\dot{\kappa }}\Delta \eta _2\Vert _{L^2_T H^\delta _x}^2&\lesssim \int \limits ^T_0 \Vert {\dot{\kappa }}\Vert _{ H^{\delta +1}_x}^2 \Vert \Delta \eta _2\Vert _{ L^2_x}^2\textrm{dt}, \\ \Vert \dot{u}\cdot \nabla \eta _2\Vert _{L^2_T H^{\delta }_x}^2&\lesssim \int \limits ^T_0 \Bigl ( \Vert \dot{u}\Vert _{ H^\delta _x}^2 \Vert \nabla \eta _2\Vert _{H^1_x}^2 +\Vert \nabla \dot{u}\Vert _{H^{\delta -\frac{1}{2}}_x}^2\Vert \nabla \eta _2\Vert _{H^{\frac{1}{2}}_x}^2\Bigr ) \textrm{dt}. \end{aligned}$$

    By use of interpolation inequalities, Gagliardo-Nirenberg’s inequalities, Young’s inequalities and Hölder’s inequalities, we derive

    $$\begin{aligned} \begin{aligned}&\Vert {\dot{\eta }}\Vert _{L^\infty _T H^{\delta +1}_x}^2+\Vert \nabla {\dot{\eta }}\Vert _{L^2_T H^{\delta +1}_x}^2\leqslant C(\delta , \kappa _*,\Vert a\Vert _{C^2}, \Vert (\theta _1,\theta _2)\Vert _{L^\infty _T H^1_x})\\&\quad \times \int \limits ^T_0 (\Vert \dot{u}\Vert _{H^{\delta }_x}^2+\Vert {\dot{\eta }}\Vert _{H^{\delta +1}_x}^2)B(t)\textrm{dt}+\frac{1}{2}\int \limits _0^T\Vert \nabla \dot{u}\Vert _{H^{\delta }_x}^2\textrm{dt}. \end{aligned} \end{aligned}$$

  • Step 2. Similarly as Step 1, we derive the following estimate

    $$\begin{aligned} \begin{aligned}&\Vert \dot{u}\Vert _{L^\infty _T H^{\delta }_x}^2+\Vert \nabla \dot{u}\Vert _{L^2_T H^{\delta }_x}^2 \\&\quad {\leqslant }C(\delta , \mu _*,\Vert b\Vert _{C^2}, \Vert (\theta _1,\theta _2)\Vert _{L^\infty _T H^1_x}) \int \limits _0^T(\Vert {\dot{\eta }}\Vert _{H^{\delta +1}_x}^2+\Vert \dot{u}\Vert _{H^{\delta }_x}^2)B(t)\textrm{dt}. \end{aligned} \end{aligned}$$

  • Step 3. We sum the above two estimates up, to derive

    $$\begin{aligned} \begin{aligned}&\Vert {\dot{\eta }}\Vert _{L^\infty _T H^{\delta +1}_x}^2+ \Vert \nabla {\dot{\eta }}\Vert _{L^2_T H^{\delta +1}_x}^2 + \Vert \dot{u}\Vert _{L^\infty _T H^{\delta }_x}^2+ \Vert \nabla \dot{u}\Vert _{L^2_T H^{\delta }_x}^2 \\&\quad {\leqslant }C(\delta , \kappa _*, \mu _*, \Vert (a,b)\Vert _{C^2}, \Vert (\theta _1,\theta _2)\Vert _{L^\infty _T H^1_x}) \int \limits _0^T(\Vert {\dot{\eta }}\Vert _{H^{\delta +1}_x}^2+\Vert \dot{u}\Vert _{H^{\delta }_x}^2)B(t)\textrm{dt}. \end{aligned} \end{aligned}$$

  Finally, the Gronwall’s inequality implies ${\dot{\eta }}=0$ and $\dot{u}=0$. The uniqueness result follows. We remark here that by virtue of the definition of B(t) we do not expect the uniqueness result below the regularity assumption $(\theta _0, u_0)\in H^1({\mathbb {R}}^2)\times (L^2({\mathbb {R}}^2))^2$, which is critical by view of the Navier–Stokes-type equation for u and the temperature-dependent diffusion coefficients.