JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION OF THE PRIMITIVE EQUATIONS IN ANISOTROPIC SPACE $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$

We consider the primitive equations

$\displaystyle\begin{array}[]{rclcl}\partial_{t}v-\Delta v+v\cdot\nabla_{H}v+w\partial_{3}v+\nabla_{H}\pi&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ \partial_{3}\pi&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ \mathrm{div}_{H}\,v+\partial_{3}w&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ v(0)&=&v_{0}&\mathrm{in}&\mathbb{T}^{3},\\ \end{array}$

(1.5)

where $u=(v,w)\in\mathbb{R}^{2}\times\mathbb{R}$ is a vector field, $\pi$ is a scalar function, $\nabla_{H}=(\partial_{1},\partial_{2})^{T}$ is the horizontal gradient, and $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ is the flat torus. We impose the periodic boundary conditions. The second equation in (1.5) is called the hydrostatic approximation. The vertical component $w$ is give by the formula

$\displaystyle w(t,x^{\prime},x_{3})=-\int_{-\pi}^{x_{3}}\mathrm{div}_{H}\,v(t,x^{\prime},\zeta)dz,\quad x=(x^{\prime},x_{3})\in\mathbb{T}^{2}\times\mathbb{T},\,t>0,$

(1.6)

where $\mathrm{div}_{H}=\nabla_{H}\cdot$ is the horizontal divergence. This formula is from the divergence-free condition. We also impose $w(\cdot,\cdot,\pm\pi)=0$. We always assume this assumption to the horizontal component of three-dimensional divergence-free vector fields in this paper. In the case of the Neumann boundary conditions in the domain $\mathbb{T}^{2}\times(-\pi,0)$ are reduced to the periodic case in $\mathbb{T}^{3}$ by the even and odd extension for $v$ and $w$, respectively. We invoke that $w$ satisfies the nonlinear parabolic equation

$\displaystyle\begin{split}\partial_{t}w-\Delta w&=\int_{-\pi}^{x_{3}}\mathrm{div}_{H}\,\left(\tilde{v}\cdot\nabla_{H}\tilde{v}\right)dz+\mathrm{div}_{H}\left(\int_{-\pi}^{x_{3}}\tilde{v}dz\cdot\nabla_{H}\overline{v}\right)\\ &+\mathrm{div}_{H}\,\left(\overline{v}\cdot\nabla_{H}\int_{-\pi}^{x_{3}}\tilde{v}dz\right)+\mathrm{div}_{H}(w\tilde{v})\\ &+\mathrm{div}_{H}\left(\int_{-\pi}^{x_{3}}(\mathrm{div}_{H}\tilde{v})\tilde{v}dz\right)\\ &-\frac{1}{2}(x_{3}-\pi)\mathrm{div}_{H}\int_{-\pi}^{x_{3}}\tilde{v}\cdot\nabla_{H}\tilde{v}+(\mathrm{div}_{H}\,\tilde{v})\tilde{v}dz\\ &=:F(v,w),\end{split}$

(1.7)

for $x_{3}\in(-\pi,\pi)$, where $\overline{v}=\frac{1}{2\pi}\int_{-\pi}^{\pi}v\,dz$ and $\tilde{v}=v-\overline{v}$. Note that (1.7) differs from the equation for $w$ derived in Proposition 4.6 of [3] in that no vertical derivatives appear in the right-hand-side terms. The derivation is described in Appendix A.

Existence of the global weak solution to the primitive equations on spherical shells was proved by Lions, Temam and Wang [13]. Local well-posedness was proved by Guillén-González, Masmoudi and Rodríguez-Bellido [10] for $H^{1}$ initial data, where $H^{s}$ is the Sobolev space with $s\in\mathbb{R}$. Cao and Titi [2] proved a $H^{1}$ energy bound to establish the global well-posedness. Hieber and Kashiwabara [11] extended this result and proved the global well-posedness in Lebesgue spaces $L^{p}$-settings for $p\geq 3$. Recently, Giga, Gries, Hieber, Hussein, and Kashiwabara [5] showed the global well-posedness in $L^{p}$-$L^{q}$ settings under the periodic, Neumann, Dirichlet, Dirichlet-Neumann mixed boundary conditions. Giga, Gries, Hieber, Hussein, and Kashiwabara [8] showed the global well-posedness in $L^{\infty}_{H}L^{1}_{x_{3}}(\mathbb{T}^{3})$, where $L^{\infty}_{H}L^{p}_{x_{3}}(\mathbb{T}^{3})$ for $q\geq 1$ denotes an anisotropic space equipped with the norm

$\displaystyle\|f\|_{L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})}:=\sup_{x^{\prime}\in\mathbb{T}^{2}}\left(\int_{\mathbb{T}}\left|f(x^{\prime},x_{3})\right|^{q}dx_{3}\right)^{1/q}.$

(1.8)

Giga, Gries, Hieber, Hussein, and Kashiwabara [7] also proved the global well-posedness in $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{2}\times(-h,0))$ for $h>0$ and $q\geq 3$ under the Dirichlet-Neumann mixed boundary conditions. An advantage of $L^{\infty}_{H}L^{q}_{x_{3}}$-approach is that one need not assume smoothness for initial data.

The aim of this paper is to give a mathematically rigorous justification of the hydrostatic approximation for the primitive equations under less smoothness assumptions than the previous works. We first introduce a brief derivation of the primitive equations. The primitive equation is derived by the Navier-Stokes equations with anisotropic viscosity, which are horizontally $O(1)$ and vertically $O(\varepsilon^{2})$. Applying a scaling to equations, we obtain the scaled Navier-Stokes equations

$\displaystyle\left.\begin{array}[]{rclcl}\partial_{t}v_{\varepsilon}-\Delta v_{\varepsilon}+u_{\varepsilon}\cdot\nabla v_{\varepsilon}+\nabla_{H}\pi_{\varepsilon}&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ \varepsilon\left(\partial_{t}w_{\varepsilon}-\Delta w_{\varepsilon}+u_{\varepsilon}\cdot\nabla w_{\varepsilon}\right)+\partial_{3}\pi_{\varepsilon}/\varepsilon&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ \mathrm{div}\,u&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ u_{\varepsilon}(0)&=&u_{0}&\mathrm{in}&\mathbb{T}^{3},\\ \end{array}\right.$

(1.13)

see [3], [4], and [12] for the details. If we multiply $\varepsilon$ to the seconde equation of (1.13) and take formal limit $\varepsilon\rightarrow 0$, then we obtain (1.5).

To justify this formal derivation we have to show the difference between the solutions to (1.5) and (1.13) converges to zero in some topologies. We put

$\displaystyle\begin{split}&U_{\varepsilon}=(V_{\varepsilon},W_{\varepsilon}),\\ &V_{\varepsilon}=v_{\varepsilon}-v,\quad W_{\varepsilon}=w_{\varepsilon}-w,\quad\Pi_{\varepsilon}=\pi_{\varepsilon}-\pi.\end{split}$

Then we see that $(U_{\varepsilon},\Pi_{\varepsilon})$ satisfies

$\displaystyle\left.\begin{array}[]{rclcl}\partial_{t}V_{\varepsilon}-\Delta V_{\varepsilon}+\nabla_{H}\pi_{\varepsilon}&=&F_{H}(U_{\varepsilon},u)&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ \varepsilon\left(\partial_{t}W_{\varepsilon}-\Delta W_{\varepsilon}\right)+\partial_{3}\Pi_{\varepsilon}/\varepsilon&=&\varepsilon F_{3}(U_{\varepsilon},u)+\varepsilon\tilde{F}(v,w)&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ \mathrm{div}\,U_{\varepsilon}&=&0&\mathrm{in}&\mathbb{T}^{3}\times(0,\infty),\\ U_{\varepsilon}(0)&=&0&\mathrm{in}&\mathbb{T}^{3},\end{array}\right.$

(1.18)

where

$\displaystyle\begin{split}F_{H}(U_{\varepsilon},u)&=-\left(U_{\varepsilon}\cdot\nabla V_{\varepsilon}+u\cdot\nabla V_{\varepsilon}+U_{\varepsilon}\cdot\nabla v\right),\\ F_{3}(U_{\varepsilon},u)&=-\left(U_{\varepsilon}\cdot\nabla W_{\varepsilon}+u\cdot\nabla W_{\varepsilon}+U_{\varepsilon}\cdot\nabla w\right),\\ \tilde{F}(v,w)&=-\left(F(v,w)+u\cdot\nabla w\right).\end{split}$

(1.19)

Note that

$\displaystyle\mathrm{div}\,U_{\varepsilon}=\mathrm{div}_{H}\,V_{\varepsilon}+\frac{\partial_{3}}{\varepsilon}(\varepsilon W_{\varepsilon})=\mathrm{div}_{\varepsilon}(V_{\varepsilon},\varepsilon W_{\varepsilon})^{T},$

where $\mathrm{div}_{\varepsilon}\,f=\mathrm{div}_{H}f^{\prime}+\partial_{3}f_{3}/\varepsilon$ for a vector field $f=(f^{\prime},f_{3})^{T}$.

The justification of the hydrostatic approximation is reduced to showing that $U_{\varepsilon}$ converges to zero. Azérad and Guillén [1] proved the weak convergence in the energy space. Li and Titi [12] showed the strong convergence in the energy space. They also proved global well-posedness to (1.13) for small $\varepsilon$ compared to the initial data. The authors together with Giga, Hieber, Hussein, and Wrona [3] extended Li and Titi’s result to $L^{p}$-$L^{q}$ settings under the Neumann boundary conditions. The authors together with Giga [4] showed the strong convergence in $L^{p}$-$L^{q}$ settings under the Dirichlet boundary conditions.

We consider the solution to (1.18) in the sense of a mild solution, namely

$\displaystyle\left(\begin{array}[]{c}V_{\varepsilon}(t)\\ \varepsilon W_{\varepsilon}(t)\end{array}\right)=\int_{0}^{t}e^{(t-s)\Delta}\mathbb{P}_{\varepsilon}\left(\begin{array}[]{c}F_{H}(U_{\varepsilon}(s),u(s))\\ \varepsilon F_{3}(U_{\varepsilon}(s),u(s))+\varepsilon\tilde{F}(v(s),w(s))\end{array}\right)ds,$

(1.24)

where $\mathbb{P}_{\varepsilon}$ is the anisotropic Helmholtz projection which maps from $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$-vector fields to $\mathrm{div}_{\varepsilon}$-free $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$-vector fields.

The first main result of this paper is the global well-posedness to (1.18) in $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$ setting for small $\varepsilon$ compared to the initial data of the primitive equations. We write $C_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})=C(\mathbb{T};L^{q}(\mathbb{T}^{2}))$ equipped with the norm (1.8) and $C_{t}C_{H}L^{q}_{x_{3}}(\mathbb{T}^{3}\times I)=C(I;C_{H}L^{q}_{x_{3}}(\mathbb{T}^{3}))$ equipped with the norm

$\displaystyle\|f\|_{C_{t}L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3}\times I)}:=\sup_{t\in I}\sup_{x^{\prime}\in\mathbb{T}^{2}}\left(\int_{\mathbb{T}}\left|f(x^{\prime},x_{3})\right|^{q}dx_{3}\right)^{1/q}.$

(1.25)

for $q\geq 1$ and an interval $I$.

The global well-posedness to the primitive equations in $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$, Theorem 1.1, and the definition of $U_{\varepsilon}$ imply

The proof of Theorem 1.1 based on the contraction mapping principle. Note that the initial data of (1.24) is zero and the external force $\varepsilon\tilde{F}(v,w)$ can be small for small $\varepsilon$. To estimate the right-hand-side of (1.24), we need $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$ bound for the composite operator $e^{t\Delta}\mathbb{P}_{\varepsilon}\partial_{j}$ for $j=1,2,3$. Since the Riesz operator is not bounded in $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$, we estimate $e^{t\Delta}\mathbb{P}_{\varepsilon}\partial_{j}$ by direct calculations for their kernel. The formula (2.10) is a key observation. To estimate $\tilde{F}(v,w)$ we need a control of $\nabla_{H}v$ since $\tilde{F}(v,w)$ has a term such as $\nabla_{H}v_{j}\nabla_{H}v_{k}$ for $j=1,2.3$ and $k=1,2$. In the semi-group approach, such kind of term cannot be estimated without additional regularity assumptions. Note that if $\nabla_{H}v_{0}\in L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$ then $\nabla_{H}v(t)\in L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$ for $t\in[0,T)$ by proof of [8] and the integral formulation for the primitive equations. For this reason we assume $\nabla_{H}v_{0}\in L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$.

To prove our main theorem, we first show short time well-posedness to (1.18) and obtain the estimate (1.1) in a short interval $[0,T_{0}]$ for $T_{0}>0$. We can extended this solution to the interval $[0,2T_{0})$ since $\|(V_{\varepsilon}(T_{0}),\varepsilon W_{\varepsilon}(T_{0}))\|_{L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})}$ is also small. Since $T$ is finite, we can establish global well-posedness to (1.18) for small $\varepsilon$. This argument is used in [3] and [4].

We introduce some notation in this paper. For $1\leq q\leq\infty$, a domain $\Omega$, and $x\in\Omega$, we write $L^{q}(\Omega)=L^{q}_{x}(\Omega)$ to denote the Lebesgue space equipped with the norm

$\displaystyle\|f\|_{L^{q}(\Omega)}=\|f\|_{L^{q}_{x}(\Omega)}:=\left(\int_{\Omega}\left|f(x)\right|^{q}dx\right)^{1/q}.$

We use the usual modification when $q=\infty$. For $1\leq q,r\leq\infty$, domains $\Omega$ and $\Omega^{\prime}$, and $(x^{\prime},x)\in\Omega^{\prime}\times\Omega$, we write $L^{q}_{x^{\prime}}L^{r}_{x}(\Omega^{\prime}\times\Omega)$ to denote anisotropic Lebesgue spaces equipped with the norm

$\displaystyle\|f\|_{L^{q}_{x^{\prime}}L^{r}_{x}(\Omega^{\prime}\times\Omega)}:=\left(\int_{\Omega^{\prime}}\left\|f(x^{\prime},\cdot)\right\|_{L^{r}(\Omega)}^{q}dx^{\prime}\right)^{1/q}.$

If $x^{\prime}$ and $x_{3}$ are the horizontal and vertical variable, respectively, then we write $L^{q}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$ to denote $L^{q}_{x^{\prime}}L^{q}_{x_{3}}(\mathbb{T}^{2}\times\mathbb{T})$. For vector fields $f$ and $g$, we denote their tensor product by $f\otimes g=(f_{i}g_{j})_{ij}$. For an integrable function $f$ on $\mathbb{T}^{3}$, we denote its vertical average by $\overline{f}=\frac{1}{2}\int_{\mathbb{T}}f(x^{\prime},z)dz$. We write $\mathcal{F}f=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)dx/(2\pi)^{d/2}$ and $\mathcal{F}^{-1}f=\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}f(\xi)d\xi/(2\pi)^{d/2}$ to denote the Fourier and Fourier inverse transform for a integrable function $f$, respectively. We denote by $\Delta_{H}=\partial_{1}^{2}+\partial_{2}^{2}$ the horizontal Laplace operator.

This paper is organized as follows. In Section 2, we show linear estimate for the heat semi-group $e^{t\Delta}$ and the composite operator $e^{t\Delta}\mathbb{P}_{\varepsilon}\partial_{j}$. We also show some estimates for composite operators having fractional derivatives. In Section 3 and Section 4, we show non-linear estimates and the external force $\tilde{F}$, respectively, from the linear estimates of Section 2. In Section 5, we prove our main theorem.

We give $L^{\infty}_{H}L^{p}_{x_{3}}$-$L^{\infty}_{H}L^{q}_{x_{3}}$-estimate for the hear semi-group. The reader refers to Section 4 of Grafakos’s book [9] for properties of the heat semi-group on $\mathbb{T}^{d}$. Let $K_{t}$ be the heat kernel on $\mathbb{T}^{d}$ for $d\geq 1$ such that

$\displaystyle K_{t}(x)=\sum_{k\in\mathbb{Z}^{d}}g_{t}(x-k),\quad x\in\mathbb{T}^{d},$

where $g$ is the Gaussian of the form

$\displaystyle g_{t}(x)=\frac{1}{\left(4\pi t\right)^{\frac{d}{2}}}e^{-\frac{|x|^{2}}{4t}},\quad x\in\mathbb{R}^{d}.$

For a integrable function $f$, we denote by $e^{t\Delta}f=K_{t}\ast f$ the heat semi-group on $\mathbb{T}^{d}$. It is known that

	$\displaystyle\|K_{t}\|_{L^{1}(\mathbb{T}^{d})}=1,\quad\|K_{t}\|_{L^{\infty}(\mathbb{T}^{d})}\leq C\left(1+\frac{1}{\sqrt{t}}\right)^{d},$
	$\displaystyle\|\partial^{\alpha}_{x}K_{t}\|_{L^{1}(\mathbb{T}^{d})}\leq Ct^{-\frac{|\alpha|}{2}},\quad\|\partial^{\alpha}_{x}K_{t}\|_{L^{\infty}(\mathbb{T}^{d})}\leq Ct^{-\frac{|\alpha|}{2}-\frac{d}{2}},$

for any multi-index $\alpha$ and some constant $C>0$, see [9] for the proof. We find from interpolation inequalities that

$\displaystyle\begin{split}&\|K_{t}\|_{L^{p}(\mathbb{T}^{d})}\leq C\left(1+\frac{1}{\sqrt{t}}\right)^{d\left(1-\frac{1}{p}\right)},\\ &\|\partial_{x}^{\alpha}K_{t}\|_{L^{p}(\mathbb{T}^{d})}\leq Ct^{-\frac{|\alpha|}{2}-\frac{d}{2}\left(1-\frac{1}{p}\right)},\end{split}$

(2.1)

for all $1\leq p\leq\infty$.

We next consider the composite operator with fraction derivative $(-\Delta)^{s/2}e^{t\Delta}f=M_{t}\ast f$ for $s>0$ and a integrable function $f$ on $\mathbb{T}^{d}$. We write $\widetilde{M}_{t}$ to denote the kernels of the corresponding composite operator on $\mathbb{R}^{d}$, respectively, namely

$\displaystyle\widetilde{M}_{t}=\mathcal{F}^{-1}|\xi|^{s}e^{-t|\xi|^{2}},\quad\xi\in\mathbb{R}^{d}.$

We know from Proposition 4.2 of Giga et al. [8] that

$\displaystyle\|\widetilde{M}_{t}\|_{L^{1}(\mathbb{R}^{d})}\leq Ct^{-\frac{s}{2}},\quad\|\widetilde{M}_{t}\|_{L^{\infty}(\mathbb{R}^{d})}\leq Ct^{-\frac{s}{2}-d}$

for some constant $C>0$. Thus we obtain

$\displaystyle\|\widetilde{M}_{t}\|_{L^{p}(\mathbb{R}^{d})}\leq Ct^{-\frac{s}{2}-\frac{d}{2}\left(1-\frac{1}{p}\right)}$

for all $1\leq p\leq\infty$. The above observations and the Poisson summation formula yield

$\displaystyle\begin{split}\|M_{t}\|_{L^{p}(\mathbb{T}^{d})}&=\left\|\frac{1}{(2\pi)^{d}}\sum_{n\in\mathbb{Z}^{d}}|n|^{s/2}e^{i\xi\cdot n}e^{-t|n|^{2}}\right\|_{L^{p}(\mathbb{T}^{d})}\\ &=\|\widetilde{M}_{t}\|_{L^{p}(\mathbb{R}^{d})}\\ &\leq Ct^{-\frac{s}{2}-\frac{d}{2}\left(1-\frac{1}{p}\right)}.\end{split}$

(2.2)

The same way as above we define the horizontal and vertical composite operators $(-\Delta_{H})^{s/2}e^{t\Delta_{H}}$ and $\partial^{s}_{3}e^{t\partial_{3}^{2}}$. It worth mentioning that, in [8], they define the vertical operator using the Caputo derivative since they consider the well-posedness in the anisotropic domain $\mathbb{R}^{2}\times\mathbb{T}$. However, we do not use the Caputo derivative since we consider in $\mathbb{T}^{3}$.

Let $\mathbb{P}_{\varepsilon}$ be the anisotropic Helmholtz projection on $\mathbb{T}^{3}$ with the matrix-valued symbol

$\displaystyle\sigma(\mathbb{P}_{\varepsilon})=I_{3}-\frac{1}{|\xi_{\varepsilon}|^{2}}\left(\begin{array}[]{c}\xi_{1}\\ \xi_{2}\\ \xi_{3}/\varepsilon\end{array}\right)\otimes\left(\begin{array}[]{c}\xi_{1}\\ \xi_{2}\\ \xi_{3}/\varepsilon\end{array}\right),\quad\xi\in\mathbb{Z}^{3},$

where $\xi_{\varepsilon}=(\xi^{\prime},\xi_{3}/\varepsilon)$ and $\sigma(A)$ denotes the symbol of a multiplier operator $A$. The projection $\mathbb{P}_{\varepsilon}$ is an unbounded operator on $L^{\infty}_{H}L^{q}_{x_{3}}(\mathbb{T}^{3})$ since the Riesz operator is unbounded on $L^{\infty}(\mathbb{T}^{d})$ for all $d\geq 1$. However, the composite operators $e^{t\Delta}\mathbb{P}_{\varepsilon}\partial_{j}$ for $j=1,2,3$ and $t>0$ are bounded on $L^{\infty}_{H}L^{p}_{x_{3}}(\mathbb{T}^{3})$. We can rewrite the anisotropic Helmholtz projection as

$\displaystyle\sigma(\mathbb{P}_{\varepsilon})=\left(\begin{array}[]{cc}I_{2}&0\\ 0&0\end{array}\right)-\frac{1}{|\xi_{\varepsilon}|^{2}}\left(\begin{array}[]{cc}\xi^{\prime}\otimes\xi^{\prime}&\xi^{\prime}\xi_{3}/\varepsilon\\ {\xi^{\prime}}^{T}\xi_{3}/\varepsilon&-|\xi^{\prime}|^{2}\end{array}\right).$

(2.10)

This is a key formula to show the boundedness for $e^{t\Delta}\mathbb{P}_{\varepsilon}\partial_{j}$. We denote by $R_{j}$ the anisotropic Riesz operator with symbol $\xi_{j}/|\xi_{\varepsilon}|$ for $j=1,2,3$. We write $R^{\prime}=(R_{1},R_{2})^{T}$.

We prove elementally estimates, which is used to show $\varepsilon$-independent bounds for composite operators. It may be somewhat prolix, but we show calculation to clarify dependence of $\varepsilon$ for the estimates.