Finite-time blowup for the 3-D viscous primitive equations of oceanic and atmospheric dynamics

Abstract

In this paper, we prove that for certain class of initial data, the corresponding solutions to the 3-D viscous primitive equations blow up in finite time. Specifically, we find a special solution to simplify the 3-D systems, assuming that the pressure function $p(x,y,t)$ is a concave function. We also consider the equations on the line $x=0$, $y=0$.

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.