1 Introduction

To the best of our knowledge, many researchers devote their studies to some nonlinear partial differential equations using mathematical physics methods [1–10]. Many works have investigated the primitive equations of ocean and atmospheric dynamics since the 1990s. In [7–13], the authors have shown the global well-posedness of strong solutions to primitive systems. In [14], it was shown that for certain class of smooth initial data, the solutions of the 2-D inviscid primitive equations are blown up in finite time by looking for a self-similar solution. In [15], the blow-up for the 2-D Prandtl equation in the maximum norm of \(u_{x}\) or \(u_{xy}\) was introduced.

It is worth mentioning that it has been proved that for certain class of initial data, the corresponding solutions of the 3-D primitive equations without viscosity blow up in finite time by looking for a self-similar solution in [16]. However, we cannot find the self-similar solution to the viscous system.

In 2023, the corresponding solutions of the 2-D viscous primitive equations were proven in [17]; the main problem was to solve the viscosity term. In this paper, we study the corresponding solutions to the 3-D primitive equations with viscosity. We need not only to solve the viscosity similarly to the method described in [17] but also to consider the effect of dimensional changes.

The three-dimensional primitive equations for large-scale oceanic and atmospheric dynamics are given by the system of partial differential equations:

$$\begin{aligned}& u_{t}+uu_{x}+vu_{y}+wu_{z}+p_{x}-Rv= \nu _{H}\Delta _{H}u+\nu _{3}u_{zz}, \end{aligned}$$
(1.1)
$$\begin{aligned}& v_{t}+uv_{x}+vv_{y}+wv_{z}+p_{y}+Ru= \nu _{H}\Delta _{H}v+\nu _{3}v_{zz}, \end{aligned}$$
(1.2)
$$\begin{aligned}& p_{z}+T=0, \end{aligned}$$
(1.3)
$$\begin{aligned}& T_{t}+uT_{x}+vT_{y}+wT_{z}=Q+\kappa _{H}\Delta _{H}T+\kappa _{3}T_{zz}, \end{aligned}$$
(1.4)
$$\begin{aligned}& u_{x}+v_{y}+w_{z}=0, \end{aligned}$$
(1.5)

with the initial value \((u_{0},v_{0},T_{0})\) and the relevant geophysical boundary conditions. Here, the horizontal velocity \((u,v)\), the vertical velocity w, the temperature T, and the pressure p are the unknowns. R is the rotation parameter, \(\nu _{H}\) is the horizontal viscosity parameter, \(\nu _{3}\) is the vertical viscosity parameter, \(\kappa _{H}\) is the horizontal diffusion parameter, and \(\kappa _{3}\) is the vertical diffusion parameter. Here, \(\Delta _{H}=\partial _{xx}+\partial _{yy}\) denotes the horizontal Laplacian operator.

2 Derivation of the reduced equations

In this section, we try to construct a special solution to the 3-D primitive equations and get the expression of the term of pressure to simplify the system. Then, we restrict the reduced equation on the surface \(x=0\), \(y=0\).

For simplicity, we take \(T(x,y,z,t)=0\) and consider the simplified primitive equations without the Coriolis force

$$\begin{aligned}& u_{t}+uu_{x}+vu_{y}+wu_{z}+p_{x}=u_{zz}, \end{aligned}$$
(2.1)
$$\begin{aligned}& v_{t}+uv_{x}+vv_{y}+wv_{z}+p_{y}=v_{zz}, \end{aligned}$$
(2.2)
$$\begin{aligned}& p_{z}=0, \end{aligned}$$
(2.3)
$$\begin{aligned}& u_{x}+v_{y}+w_{z}=0, \end{aligned}$$
(2.4)

in the horizontal channel \(\Omega =\{(x,y,z):0\leq z\leq H,(x,y)\in D\subset R^{2}\}\) and \(t\in [0,T)\), with initial and boundary conditions:

$$\begin{aligned}& (u,v)|_{t=0}=(u_{0},v_{0}) (x,y,z), \end{aligned}$$
(2.5)
$$\begin{aligned}& (u,v,w)|_{z=0}=(u,v,w)|_{z=H}=0, \end{aligned}$$
(2.6)
$$\begin{aligned}& (u,v)|_{\partial D}=(u_{1},v_{1}). \end{aligned}$$
(2.7)

Specifically, we assume that \(p(x,y,t)\) is a concave function, with respect to the variable x. Then, we have

$$ p_{xx}\geq 0. $$
(2.8)

We construct the solution to the system (2.1)–(2.4) with the structure

$$ \bigl(u(x,y,z,t),k(x,y,t)u(x,y,z,t),w(x,y,z,t)\bigr), $$
(2.9)

with \(u(x,y,z,t)\) being strictly increasing in \(0 \leq z \leq H \),

$$ u_{z}>0. $$
(2.10)

After the above assumptions, we have

$$ v_{0}=ku_{0},\qquad v_{1}=ku_{1}. $$
(2.11)

Plugging the form (2.9) into (2.2) and using (2.1), we get

$$ uk_{t}+u^{2}k_{x}+ku^{2}k_{y}-k p_{x}+p_{y}=0. $$
(2.12)

By differentiating (2.12) with respect to z and using (2.10), we have

$$ k_{t}+2u(k_{x}+k k_{y})=0. $$
(2.13)

By differentiating (2.13) with respect to z, we get

$$ (k_{x}+k k_{y})=0. $$
(2.14)

Combing (2.13) and (2.14), we get \(k_{t}=0\). Plugging these equations into (2.12) gives the following

$$ p_{y}-k p_{x}=0. $$
(2.15)

The systems to be studied in this paper can be formulated as follows

$$\begin{aligned}& u_{t}+uu_{x}+vu_{y}+wu_{z}+p_{x}=u_{zz}, \end{aligned}$$
(2.16)
$$\begin{aligned}& v_{t}+uv_{x}+vv_{y}+wv_{z}+p_{y}=v_{zz}, \end{aligned}$$
(2.17)
$$\begin{aligned}& u_{x}+v_{y}+w_{z}=0, \end{aligned}$$
(2.18)

with initial and boundary conditions:

$$\begin{aligned}& (u,v)|_{t=0}=(u_{0},ku_{0}) (x,y,z), \end{aligned}$$
(2.19)
$$\begin{aligned}& (u,v,w)|_{z=0}=(u,v,w)|_{z=H}=0, \end{aligned}$$
(2.20)
$$\begin{aligned}& (u,v)|_{\partial D}=(u_{1},ku_{1}), \end{aligned}$$
(2.21)

there \(u_{z}(x,y)\), \(k(x,y)\) and \(p(x,y,t)\) satisfy (2.10), (2.14), and (2.15).

Lemma 2.1

If \((u,v,w)(x,y,z,t)\) is the classical solution to (2.16)–(2.21), then \(v(x,y,z,t)=k(x,y)u(x,y,z,t)\). See Lemma 2.1 in [16].

Therefore, studying the problem (2.16)–(2.21) is equivalent to studying the following reduced problem for only two unknown functions u and w

$$\begin{aligned}& u_{t}+uu_{x}+ku\cdot u_{y}+wu_{z}+p_{x}=u_{zz}, \end{aligned}$$
(2.22)
$$\begin{aligned}& u_{x}+(ku)_{y}+w_{z}=0, \end{aligned}$$
(2.23)

with initial and boundary condition

$$\begin{aligned}& (u,w)|_{z=0}=(u,w)|_{z=H}=0, \end{aligned}$$
(2.24)
$$\begin{aligned}& u|_{\partial D}=u_{1}, \end{aligned}$$
(2.25)
$$\begin{aligned}& u|_{t=0}=u_{0}(x,y,z). \end{aligned}$$
(2.26)

In addition, we impose the following condition

$$\begin{aligned}& \begin{aligned} & u(x,y,z,t)=-u(-x,y,z,t);\qquad w(x,y,z,t)=w(-x,y,z,t);\qquad \\ & p(x,y,t)=p(-x,y,z,t). \end{aligned} \end{aligned}$$
(2.27)

By differentiating equation (2.22) with respect to x, we obtain

$$ u_{tx}+u^{2}_{x}+uu_{xx}+(ku)_{x}u_{y}+ku \cdot u_{xy}+w_{x}u_{z}+wu_{xz}+p_{xx}=u_{zzx}. $$
(2.28)

By averaging (2.28) with respect to the z variable over \([0,H]\) and multiplying \(\frac{1}{H}\), we obtain

$$ \frac{1}{H} \int ^{H}_{o}\bigl[u_{tx}+2u^{2}_{x}+2uu_{xx}+(ku \cdot u)_{xy}-u_{zzx}\bigr]\,dz+p_{xx}=0. $$
(2.29)

Thus,

$$ \frac{1}{H} \int ^{H}_{o}\bigl[u_{t}+2uu_{x}+(ku \cdot u)_{y}-u_{zz}\bigr]\,dz+p_{x}=C(y,t) $$
(2.30)

for some function \(C(y,t)\). Due to (2.28), we know that \(p_{x}\) and u are odd functions, with respect to the variable x, then

$$ C(y,t)=0, $$
(2.31)

and consequently

$$ p_{x}=-\frac{1}{H} \int ^{H}_{0}\bigl[u_{t}+2uu_{x}+(ku \cdot u)_{y}-u_{zz}\bigr]\,dz. $$
(2.32)

Substituting (2.32) into system (2.22), we obtain the closed system

$$ u_{t}+uu_{x}+ku\cdot u_{y}+wu_{z}-u_{zz}=- \frac{1}{H} \int ^{H}_{0}\bigl[u_{t}+2uu_{x}+(ku \cdot u)_{y}-u_{zz}\bigr]\,dz. $$
(2.33)

By differentiating with respect to x, we have

$$\begin{aligned}& u_{tx}+u^{2}_{x}+uu_{x}+(ku)_{x}u_{y}+ku \cdot u_{xy}+w_{x}u_{z}+wu_{xz}-u_{zzx} \\& \quad = \frac{1}{H} \int ^{H}_{0}\bigl[u_{tx}+2u^{2}_{x}+2uu_{xx}+(ku \cdot u)_{yx}-u_{xzz}\bigr]\,dz. \end{aligned}$$
(2.34)

Let us consider the restriction of the evolution of equation (2.34) on the surface \(x=0\), \(y=0\). Since u is an odd function, and w is an even function, with respect to the variable x, we have

$$ u(0,0,z,t)=0; \qquad w_{x}(0,0,z,t)=0. $$
(2.35)

This, together with (2.34), implies

$$\begin{aligned}& u_{tx}(0,0,z,t)+\bigl(u_{x}(0,0,z,t)\bigr)^{2}- \int ^{z}_{0}u_{x}\,dz\cdot u_{xz}(0,0,z,t) \\& \quad =\frac{1}{H} \int ^{H}_{0}\bigl[u_{tx}(0,0,z,t)+2 \bigl(u_{x}(0,0,z,t)\bigr)^{2}-u_{xzz}\bigr]\,dz. \end{aligned}$$
(2.36)

Denoting by

$$ a(z,t)=-u_{x}(0,0,z,t), $$
(2.37)

we obtain

$$ a_{t}=a^{2}+a_{zz}- \int ^{z}_{0}a \,dz\cdot a_{z}- \frac{1}{H} \int ^{H}_{0}\bigl(-a_{t}+2a^{2}+a_{zz} \bigr)\,dz, $$
(2.38)

with the initial and boundary conditions

$$ a(z,0)=a_{0}(z), \qquad a(0,t)=0, \qquad a(H,t)=0. $$
(2.39)

In particular, due to equations (2.8) and (2.32), we have

$$ f(t)=\frac{1}{H} \int ^{H}_{0}\bigl(-a_{t}+2a^{2}+a_{zz} \bigr)\,dz\leq 0. $$
(2.40)

3 Main result and the proof

In this section, we will give Lemma 3.1 and the main result.

Lemma 3.1

Define

$$ F(a)= \int ^{H}_{0}a^{2}\,dz , \qquad E(a)= \int ^{H}_{0}\biggl(\frac{1}{2}a^{2}_{z}- \frac{1}{4}a^{3}\biggr)\,dz, \qquad G=-\frac{E}{F^{\beta}}, $$
(3.1)

where \(\beta \in (1,\frac{5}{4})\). Let \(a_{0}\) be nonnegative, compactly supported initial data such that \(E(a_{0})<0\), \(a_{zz}(0,t)=0\) and \(a_{zz}(H,t)=0\). Then, there exists a finite time T such that either

$$ \lim_{t\rightarrow T}\max_{z}a=+\infty , $$
(3.2)

or at least one of the following results

$$\begin{aligned}& \lim_{t\rightarrow T}a_{z}(0,t)=+\infty , \end{aligned}$$
(3.3)
$$\begin{aligned}& \lim_{t\rightarrow T}a_{z}(H,t)=+\infty . \end{aligned}$$
(3.4)

Remark 3.1

Using Lemma 2.1, we reduce the system (2.1)–(2.7) to the system (2.22)–(2.26) with two unknown functions and three variables. Then, we consider the problem on the surface \(x=0\) and \(y=0\). So, the proof of Lemma 3.1 takes the same method as in [17].

Theorem 3.1

Assuming that \(p_{xx}\geq 0\) and the conditions (2.10) and (2.27) are satisfied, we can get Lemma 3.1. Then, smooth solutions to (2.16)–(2.21) do not exist globally in time.

Remark 3.2

In the Lemma 3.1, \(a(z,t)=-u_{x}(0,0,z,t)\), so the blowup is in the norm of \(u_{x}\) or \(u_{xy}\).

4 Conclusions

This study has shown that the 3-D primitive equations with viscosity blow up on the surface \(x=0\) and \(y=0\). In this paper, we consider the simplified equations. We plan to solve unreduced systems in the future.