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

*Boundary Value Problems*
**volumeÂ 2024**, ArticleÂ number:Â 54 (2024)

## 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 methodsÂ [1â€“10]. Many works have investigated the primitive equations of ocean and atmospheric dynamics since the 1990s. In [7â€“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:

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 pressureÂ *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

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:

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

We construct the solution to the system (2.1)â€“(2.4) with the structure

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

After the above assumptions, we have

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

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

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

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

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

with initial and boundary conditions:

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)*â€“*(2.21), *then* \(v(x,y,z,t)=k(x,y)u(x,y,z,t)\). *See LemmaÂ *2.1 *in* [16].

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

with initial and boundary condition

In addition, we impose the following condition

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

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

Thus,

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

and consequently

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

By differentiating with respect to *x*, we have

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

This, together with (2.34), implies

Denoting by

we obtain

with the initial and boundary conditions

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

## 3 Main result and the proof

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

### Lemma 3.1

*Define*

*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*

*or at least one of the following results*

### Remark 3.1

Using LemmaÂ 2.1, we reduce the system (2.1)â€“(2.7) to the system (2.22)â€“(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 LemmaÂ 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 LemmaÂ *3.1. *Then*, *smooth solutions to* (2.16)*â€“*(2.21) *do not exist globally in time*.

### Remark 3.2

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

## 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.

## Data Availability

No datasets were generated or analysed during the current study.

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Lin Zheng wrote the whole manuscript and reviewed it.

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### Cite this article

Zheng, L. Finite-time blowup for the 3-D viscous primitive equations of oceanic and atmospheric dynamics.
*Bound Value Probl* **2024**, 54 (2024). https://doi.org/10.1186/s13661-024-01858-y

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DOI: https://doi.org/10.1186/s13661-024-01858-y