Computability of the Solutions to Navier-Stokes Equations via Effective Approximation

Abstract

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics, Pour-El and Richards proposed “... the recursion theoretic study of particular nonlinear problems of classical importance. Examples are the Navier-Stokes equation, the KdV equation, and the complex of problems associated with Feigenbaum’s constant.” In this paper, we approach the question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch’s Type-2 Theory of Effectivity. A natural encoding (“representation”) is constructed for the space of divergence-free vector fields on 2-dimensional open square $\varOmega = (-1, 1)^2$. This representation is shown to render first the mild solution to the Stokes Dirichlet problem and then a strong local solution to the nonlinear inhomogeneous incompressible Navier-Stokes initial value problem uniformly computable. Based on classical approaches, the proofs make use of many subtle and intricate estimates which are developed in the paper for establishing the computability results.

The paper is dedicated to the memory of Professor Ker-I Ko.

The third author is supported by grant NRF-2017R1E1A1A03071032.

Appendices

A Proof of Proposition 1

(a) For a divergence-free and boundary-free polynomial, its coefficients must satisfy a system of linear equations. In the following, we derive explicitly this system of linear equations in the 2-dimensional case, i.e. $\varOmega = (-1, 1)^2$. Let $\varvec{p}=(p_1, p_2)=\big (\sum _{i, j=0}^{N}a^1_{i, j}x^iy^j, \sum _{i, j=0}^{N}a^2_{i, j}x^iy^j\big )$ be a divergence-free and boundary-free polynomial of real coefficients. (If the degree of $p_1$ or $p_2$ is less than N, then zeros are placed for the coefficients of missing terms). Then, by definition,

$$\begin{aligned} \nabla \cdot \varvec{p}= & {} \frac{\partial p_1}{\partial x}+\frac{\partial p_2}{\partial y} \\= & {} \sum _{1\le i\le N, 0\le \le N}ia^1_{i, j}x^{i-1}y^j + \sum _{0\le i\le N, 1\le \le N}ja^2_{i, j}x^{i}y^{j-1} \\= & {} \sum _{0\le i, j\le N-1}[(i+1)a^1_{i+1, j}+(j+1)a^2_{i, j+1}]x^{i}y^j \\&+ \sum _{0\le i\le N-1}(i+1)a^1_{i+1, N}x^iy^N + \sum _{0\le j\le N-1}(j+1)a^2_{N, j+1}x^Ny^j \\\equiv & {} 0 \quad \hbox {on} \, \varOmega \end{aligned}$$

which implies that all coefficients in $\nabla \cdot \varvec{p}$ must be zero; or equivalently, Eq. (7) holds true. Turning to the boundary conditions, along the line $x=1$, since

$$\begin{aligned} \varvec{p} (1, y) = \big (\sum \nolimits _{j=0}^N(\sum \nolimits _{i=0}^N a^1_{i, j})y^j, \sum \nolimits _{j=0}^N(\sum \nolimits _{i=0}^N a^2_{i, j})y^j \big ) \end{aligned}$$

is identically zero, it follows that $\sum _{i=0}^N a^1_{i, j}=\sum _{i=0}^N a^2_{i, j}=0$ for $0\le j\le N$. There are similar types of restrictions on the coefficients of $\varvec{p}$ along the lines $x=-1$, $y=1$, and $y=-1$. In summary, $\varvec{p}$ vanishes on $\partial \varOmega$ if and only if for all $0\le j,i\le N$, both (8) and (9) hold true.

In the 3-dimensional case, a similar calculation shows that a polynomial triple $\varvec{p}(x, y, z)=\big (p_1(x, y, z), p_2(x, y, z), p_3(x, y, z)\big )$ is divergence-free and boundary-free if and only if its coefficients satisfies a system of linear equations with integer coefficients.

(b) In [8] it is shown that for any real number $s\ge 3$ and for any function $\varvec{ w}\in \mathcal {N}_{div}^s\cap H_{2,0}^{1,\sigma }(\varOmega )^d$, the following holds:

$$\begin{aligned} \inf _{\varvec{p}\in \mathcal {N}^1_{\text {div}}\bigcap \mathcal {P}_N^0(\varOmega )^d}\Vert \varvec{ w}-\varvec{p}\Vert _{H_{2}^{s}(\varOmega )^d}\le CN^{-2}\Vert \varvec{ w}\Vert _{H_{2}^{s}(\varOmega )^d} \end{aligned}$$

where $\varOmega =(-1, 1)^d$,

$$\begin{aligned} \mathcal {N}_{\text {div}}^s =\{ \varvec{ w}\in H_{2}^{s}(\varOmega )^d \, | \, \nabla \cdot \varvec{w}=0\},\quad \mathcal {P}_N^0(\varOmega )=\mathcal {P}_N(\varOmega )\bigcap H_{2,0}^{1,\sigma }(\varOmega ), \end{aligned}$$

$\mathcal {P}_N$ is the set of all d-tuples of real polynomials with d variables and degree less than or equal to N with respect to each variable, $H_{2,0}^{1,\sigma }(\varOmega )$ is the closure in $H_{2}^{1}(\varOmega )$ of $C^\infty _0(\varOmega )$, and C is a constant independent of N. This estimate implies that every function $\varvec{ w}\in L^{\sigma }_{2,0}$ can be approximated with arbitrary precision by divergence-free and boundary-free real polynomials as follows: for any $n\in \mathbb {N}$, since $\{ \varvec{ u}\in C^\infty _0(\varOmega )^d \, : \, \nabla \cdot \varvec{ u}=0\}$ is dense in $L^{\sigma }_{2,0}$, there is a divergence-free $C^\infty$ function $\varvec{ u}$ with compact support in $\varOmega$ such that $\Vert \varvec{ w}-\varvec{u}\Vert _{L_{2}}\le 2^{-(n+1)}$. Then it follows from the above inequality that there exists a positive integer N and a divergence-free and boundary-free polynomial $\varvec{p}$ of degree N with real coefficients such that $\Vert \varvec{ u} - \varvec{ p}\Vert _{L_{2}}\le \Vert \varvec{ u}-\varvec{p}\Vert _{H^3(\varOmega )^d}\le 2^{-(n+1)}$. Consequently, $\Vert \varvec{ w}-\varvec{ p}\Vert _{L_{2}}\le \Vert \varvec{ w}-\varvec{ u}\Vert _{L_{2}}+\Vert \varvec{ u}-\varvec{ p}\Vert _{L_{2}}\le 2^{-n}$.

It remains to show that $\mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]$, the divergence-free and boundary-free polynomial tuples with rational coefficients, is dense (in $L_{2}$-norm) in the set of all polynomial tuples with real coefficients which are divergence-free on $\varOmega$ and boundary-free on $\partial \varOmega$. To this end we note that, according to part (a), the divergence-free and boundary-free polynomials can be characterized, independent of their coefficient field, in terms of a homogeneous system of linear equations with integer coefficients. Then it follows from the lemma below that the set of the rational solutions of this system is dense in the set of its real solutions. And since $\varOmega$ is bounded (=relatively compact), the approximations to its coefficients of a polynomial yields (actually uniform) the approximations to the polynomial itself:

$$\begin{aligned} \sup _{\varvec{ x}\in \varOmega } |p_k(\varvec{ x})| \;\le \; \sum _{i, j=0}^{N} |a^k_{i,j}|\cdot M^{i+j} \quad \text { for } \varOmega \subseteq [-M,+M]^2 \quad \text {and } k=1,2 \end{aligned}$$

Lemma 7

Let $A\in \mathbb {Q}^{m\times n}$ be a rational matrix. Then the set $\mathrm {kernel}_{IQ}(A):=\{\varvec{ x}\in \mathbb {Q}^n:A\cdot \varvec{ x}=\varvec{ 0}\}$ of rational solutions to the homogeneous system of linear equations given by A is dense in the set $\mathrm {kernel}_{\mathbb {R}}(A)$ of real solutions.

Proof

For $d:=\mathrm {dim}\big (\mathrm {kernel}_{\mathbb {R}}(A)\big )$, Gaussian Elimination yields a basis $B=(\varvec{ b}^1,\ldots ,\varvec{ b}^d)$ of $\mathrm {kernel}(A)$; in fact it holds $B\in \mathbb {Q}^{n\times d}$ and

$$\begin{aligned} \mathrm {kernel}_{\mathbb {F}}(A) \;=\; \mathrm {image}_{\mathbb {F}}(B) \;:=\; \big \{\lambda _1\varvec{ b}^1+\cdots +\lambda _d\varvec{ b}^d:\lambda _1,\ldots ,\lambda _d\in \mathbb {F}\big \} \end{aligned}$$

for every field $\mathbb {F}\supseteq \mathbb {Q}$: Observe that the elementary row operations Gaussian Elimination employs to transform A into echelon form containing said basis B consist only of arithmetic (=field) operations! (We deliberately do not require B to be orthonormal; cf. [29, §3]). Now $\mathrm {image}_{\mathbb {Q}}(B)$ is obviously dense in $\mathrm {image}_{\mathbb {R}}(B)$.

B Proof of Lemma 1

Note that $\gamma _n*\mathbb {T}_k\varvec{p} = (\gamma _n*\mathbb {T}_kp_1, \gamma _n*\mathbb {T}_kp_2)$. For each $\varvec{p}\in \mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]$ and $n\ge k$, since

$$\begin{aligned} \frac{\partial (\gamma _n*\mathbb {T}_kp_1)}{\partial x}&(x, y) = \frac{\partial }{\partial x}\int ^{1}_{-1}\int ^{1}_{-1}\gamma _n(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)\,ds\,dt \nonumber \\&= \int ^{1-2^{-k}}_{-1+2^{-k}}\left[ \int ^{1-2^{-k}}_{-1+2^{-k}}\frac{\partial \gamma _n}{\partial x}(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)\,ds\right] \,dt \nonumber \\&= \int ^{1-2^{-k}}_{-1+2^{-k}}\left[ \int ^{1-2^{-k}}_{-1+2^{-k}}-\frac{\partial \gamma _n}{\partial s}(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)\,ds\right] \,dt \end{aligned}$$

(37)

for $\mathbb {T}_kp_1=0$ in the exterior region of $\varOmega _k$ including its boundary $\partial \varOmega _k$. Note that $\frac{\partial \gamma _n}{\partial s}$ is continuous on $\mathbb {R}^2$; $\frac{\partial \gamma _n}{\partial s}(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)$ is continuous on $[-1, 1]^2$ for any given $x, y\in \mathbb {R}$; $\frac{\partial \mathbb {T}_kp_1}{\partial s}(s, t)$ is continuous in $(-1+2^{-n}, 1-2^{-n})$ and $\mathbb {T}_kp_1$ is continuous on $[-1+2^{-n}, 1-2^{-n}]$ for any given $t\in [-1; 1]$. Thus, we can apply the integration by parts formula to the integral

$$\int ^{1-2^{-k}}_{-1+2^{-k}}-\frac{\partial \gamma _n}{\partial s}(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)\,ds$$

as follows:

$$\begin{aligned}&\quad \int ^{1-2^{-k}}_{-1+2^{-k}} -\frac{\partial \gamma _n}{\partial s}(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)\,ds \nonumber \\&\quad = -\gamma _n(x-s, y-t)\cdot \mathbb {T}_kp_1(s, t)\big | ^{1-2^{-k}}_{-1+2^{-k}} \nonumber \\&\qquad \qquad \qquad +\int ^{1-2^{-k}}_{-1+2^{-k}}\gamma _n(x-s, y-t) \cdot \frac{\partial \mathbb {T}_kp_1}{\partial s}(s, t)\,ds \nonumber \\&\quad = \int ^{1-2^{-k}}_{-1+2^{-k}}\gamma _n(x-s, y-t)\cdot \frac{\partial \mathbb {T}_kp_1}{\partial s}(s, t)\,ds \end{aligned}$$

(38)

Then it follows from (37) and (38) that for any $(x, y)\in \varOmega$,

$$\begin{aligned} \frac{\partial \gamma _n*\mathbb {T}_kp_1}{\partial x}(x, y)= \int ^{1-2^{-k}}_{-1+2^{-k}}\int ^{1-2^{-k}}_{-1+2^{-k}}\gamma _n(x-s, y-t)\cdot \frac{\partial \mathbb {T}_kp_1}{\partial s}(s, t)\,ds\,dt . \end{aligned}$$

A similar calculation yields that for any $(x, y)\in \varOmega$,

$$\begin{aligned} \frac{\partial \gamma _n*\mathbb {T}_kp_2}{\partial y}(x, y)= \int ^{1-2^{-k}}_{-1+2^{-k}}\int ^{1-2^{-k}}_{-1+2^{-k}}\gamma _n(x-s, y-t)\cdot \frac{\partial \mathbb {T}_kp_2}{\partial t}(s, t)\,ds\,dt . \end{aligned}$$

Thus, for any $(x, y)\in \varOmega$ and $n\ge k$,

$$\begin{aligned} \nabla \cdot&(\gamma _n*\mathbb {T}_k\varvec{p})(x, y) = \frac{\partial \gamma _n*\mathbb {T}_kp_1}{\partial x}(x, y)+ \frac{\partial \gamma _n*\mathbb {T}_kp_2}{\partial y}(x, y) \\&= \int ^{1-2^{-k}}_{-1+2^{-k}}\int ^{1-2^{-k}}_{-1+2^{-k}}\gamma _n(x-s, y-t)\cdot \left[ \frac{\partial \mathbb {T}_kp_1}{\partial s}+\frac{\partial \mathbb {T}_kp_2}{\partial t}\right] (s, t) \,ds\,dt \\&=0 \end{aligned}$$

for $\mathbb {T}_k\varvec{p}=(\mathbb {T}_kp_1, \mathbb {T}_kp_2)$ is divergence-free on $\varOmega _k$. This proves that for any $\varvec{p}\in \mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]$ and $n\ge k$, $\gamma _n*\mathbb {T}_k\varvec{p}$ is divergence-free on $\varOmega$.

C Proof of Lemma 2

Since for each $\varvec{p}\in \mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]$ and $k\in \mathbb {N}$, $\gamma _n*\mathbb {T}_k\varvec{p}\rightarrow \mathbb {T}_k\varvec{p}$ effectively and uniformly on $\varOmega _k$ as $n\rightarrow \infty$, it suffices to show that $\{\mathbb {T}_k\varvec{p}:k\in \mathbb {N}, \varvec{p}\in \mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]\}$ is dense in $L^{\sigma }_{2,0}(\varOmega )$. On the other hand, since $\mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]$ is dense in $L^{\sigma }_{2,0}(\varOmega )$, we only need to show that for each $\varvec{p}\in \mathbb {Q}^{\sigma }_{0}[\mathbb {R}^2]$ and $m\in \mathbb {N}$, there is a $k\in \mathbb {N}$ such that $2^{-m}\ge \Vert \varvec{p}-\mathbb {T}_k\varvec{p}\Vert _\infty = \max \{|p_1(\varvec{ x})-\mathbb {T}_kp_1(\varvec{ x})|,|p_2(\varvec{ x})-\mathbb {T}_kp_2(\varvec{ x})|:\varvec{ x}\in \bar{\varOmega }\}$.

Since $p_i$ is uniformly continuous on $\bar{\varOmega }$, there exists a $k\in \mathbb {N}$ such that $|p_i(x,y)-p_i(x',y')|\le 2^{-m}$ whenever $|x-x'|,|y-y'|\le 2^{-k+1}$, and, in particular, for $x'=\frac{x}{1-2^{-k}}$ and $y'=\frac{y}{1-2^{-k}}$. Also, since $p_i(x,y)=0$ for $(x,y)\in \partial \varOmega$, $|p_i(x,y)|\le 2^{-m-1}$ for all $(x,y)\in \varOmega \setminus \varOmega _k$. This establishes $|p_i(x,y)-\mathbb {T}_kp_i(x,y)|\le 2^{-m}$ on $\bar{\varOmega }$.