Existence and uniqueness of solutions for the two-dimensional Euler and Navier-Stokes equations with initial data in $H^1$

1.   Introduction

In this paper, we propose shifted-Legendre orthogonal function method for high-dimensional heat conduction equation ^[1]:

Where $u(t, x, y, z)$ is the temperature field, $\phi(x, y, z)$ is a known function, $k$ is the thermal diffusion efficiency, and $a, b, c$ are constants that determine the size of the space.

Heat conduction system is a very common and important system in engineering problems, such as the heat transfer process of objects, the cooling system of electronic components and so on ^[1,2,3,4]. Generally, heat conduction is a complicated process, so we can't get the analytical solution of heat conduction equation. Therefore, many scholars proposed various numerical algorithms for heat conduction equation ^[5,6,7,8]. Reproducing kernel method is also an effective numerical algorithm for solving boundary value problems including heat conduction equation ^{[9,10,11,12,13,14]}. Galerkin schemes and Green's function are also used to construct numerical algorithms for solving one-dimensional and two-dimensional heat conduction equations ^{[15,16,17,18,19]}. Alternating direction implicit (ADI) method can be very effective in solving high-dimensional heat conduction equations ^[20,21]. In addition, the novel local knot method and localized space time method are also used to solve convection-diffusion problems ^{[22,23,24,25]}. These methods play an important reference role in constructing new algorithms in this paper.

Legendre orthogonal function system is an important function sequence in the field of numerical analysis. Because its general term is polynomial, Legendre orthogonal function system has many advantages in the calculation process. Scholars use Legendre orthogonal function system to construct numerical algorithm of differential equations ^[26,27,28].

Based on the orthogonality of Legendre polynomials, we delicately construct a numerical algorithm that can be extended to high-dimensional heat conduction equation. The proposed algorithm has $\alpha$-Order convergence, and our algorithm can achieve higher accuracy compared with other algorithms.

The content of the paper is arranged like this: The properties of shifted Legendre polynomials, homogenization and spatial correlation are introduced in Section 2. In Section 3, we theoretically deduce the numerical algorithm methods of high-dimensional heat conduction equations. The convergence of the algorithm is proved in Section 4. Finally, three numerical examples and a brief summary are given at the end of this paper.

2.   Preliminaries

In this section, the concept of shifted-Legendre polynomials and the space to solve Eq (1.1) are introduced. These knowledge will pave the way for describing the algorithm in this paper.

2.1.   Shifted-Legendre polynomial

The traditional Legendre polynomial is the orthogonal function system on $[-1, 1]$. Since the variables $t, x, y, z$ to be analyzed for Eq (1.1) defined in different intervals, it is necessary to transform the Legendre polynomial on $[c_1, c_2]$, $c_1, c_2\in \mathbb{R}$, and the shifted-Legendre polynomials after translation transformation and expansion transformation by Eq (2.1).

Obviously, $\{p_i(x)\}_{i = 0}^\infty$ is a system of orthogonal functions on $L^2[c_1, c_2]$, and

Let $L_i(x) = \sqrt{\frac{2 i+1}{c_2-c_1}}p_i(x)$. Based on the knowledge of ref. [29], we begin to discuss the algorithm in this paper.

Lemma 2.1. ^[29] $\{L_i(x)\}_{i = 0}^\infty$ is a orthonormal basis in $L^2[c_1, c_2]$.

2.2.   Homogenization of boundary value conditions

Considering that the problem studied in this paper has a nonhomogeneous boundary value condition, the problem (1.1) can be homogenized by making a transformation as follows.

Here, homogenization is necessary because we can easily construct functional spaces that meet the homogenization boundary value conditions. This makes us only need to pay attention to the operator equation itself in the next research, without considering the interference caused by boundary value conditions.

In this paper, in order to avoid the disadvantages of too many symbols, the homogeneous heat conduction system is still represented by $u$, the thermal diffusion efficiency $k = 1$, and the homogeneous system of heat conduction equation is simplified as follows:

2.3.   Introduction of the space

The solution space of Eq (2.2) is a high-dimensional space, which can be generated by some one-dimensional spaces. Therefore, this section first defines the following one-dimensional space.

Remember $AC$ represents the space of absolutely continuous functions.

Definition 2.1. $W_1[0, 1] = \{u(t)|u\in AC, u(0) = 0, u'\in L^2[0, 1]\},$ and

Let $c_1 = 0, c_2 = 1$, so $\{T_i(t)\}_{i = 0}^\infty$ is the orthonormal basis in $L^2[0, 1]$, where $T_i(t) = L_i(t)$, note $T_n(t) = \sum\limits_{i = 0}^n c_i t^i$. And $\{JT_n(t)\}_{n = 0}^\infty$ is the orthonormal basis of $W_1[0, 1]$, where

Definition 2.2. $W_2[0, a] = \{u(x)|u'\in AC, u(0) = u(a) = 0, u''\in L^2[0, a]\},$ and

Similarly, $\{P_n(x)\}_{n = 0}^\infty$ is the orthonormal basis in $L^2[0, a]$, and denote $P_n(x) = \sum\limits_{j = 0}^n d_j x^j$, where $d_j\in \mathbb{R}$.

Let

obviously, $\{JP_n(x)\}_{n = 0}^\infty$ is the orthonormal basis of $W_2[0, a]$.

3.   Numerical method and analysis

We start with solving one-dimensional heat conduction equation, and then extend the algorithm to high-dimensional heat conduction equations.

3.1.   The scheme of one-dimensional model

Let $D = [0, 1]\times[0, a]$, $CC$ represents the space of completely continuous functions, and $\mathbb{N}_n$ represents a set of natural numbers not exceeding $n$.

Definition 3.1. $W(D) = \{u(t, x)|\frac{\partial u}{\partial x}\in CC, (t, x)\in D, u(0, x) = 0, u(t, 0) = u(t, a) = 0, \frac{\partial^3 u}{\partial t\partial x^2}\in L^2(D)\},$ and

Theorem 3.1. $W(D)$ is an inner product space.

Proof. $\forall u(t, x)\in W(D)$, if $\langle u, u\rangle_{W(D)} = 0$, means

and it implies

Combined with the conditions of $W(D)$, we can get $u = 0$.

Obviously, $W(D)$ satisfies other conditions of inner product space.

Theorem 3.2. $\forall u\in W(D), v_1(t)v_2(x)\in W(D)$, then

Corollary 3.1. $\forall u_1(t)u_2(x)\in W(D), v_1(t)v_2(x)\in W(D)$, then

Let

Theorem 3.3. $\{\rho_{ij}(t, x)\}_{i, j = 0}^\infty \mathit{\mbox{is an orthonormal basis in}}W(D)$.

Proof. $\forall \rho_{ij}(t, x), \rho_{lm}(t, x)\in W(D), \; \; i, j, l, m\in \mathbb{N}$,

So

In addition, $\forall u\in W(D)$, if $\langle u, \rho_{ij}\rangle_{W(D)} = 0$, means

Note that $\{JP_j(x)\}_{j = 0}^\infty$ is the complete system of $W_2$, so $\langle u(t, x), JT_i(t)\rangle_{W_1} = 0$.

Similarly, we can get $u(t, x) = 0$.

Let $\mathcal{L}: W(D) \rightarrow L^2(D)$,

So, Eq (3.1) can be simplified as

Definition 3.2. $\forall\varepsilon > 0$, if $u\in W(D)$ and

then $u$ is called the $\varepsilon-$best approximate solution for $\mathcal{L}u = f$.

Theorem 3.4. Any $\varepsilon > 0$, there is $N\in \mathbb{N}$, when $n > N$, then

is the $\varepsilon-$best approximate solution for $\mathcal{L}u = f$, where $\eta^*_{ij}$ satisfies

Proof. According to the Theorem 3.3, if $u$ satisfies Eq (3.2), then $u(t, x) = \sum\limits_{i = 0}^{\infty}\sum\limits_{j = 0}^{\infty}\eta_{ij}\rho_{ij}(t, x)$, where $\eta_{ij}$ is the Fourier coefficient of $u$.

Note that $\mathcal{L}$ is a bounded operator ^[30], hence, any $\varepsilon > 0$, there is $N\in \mathbb{N}$, when $n > N$, then

So,

For obtain $u_n(t, x)$, we need to find the coefficients $\eta_{ij}^{*}$ by solving Eq (3.5).

In addition,

So,

and the equations $\frac{\partial J}{\partial \eta_{ij}} = 0, \; \; i, j \in \mathbb{N}_n$ can be simplified to

where

Theorem 3.5. ${\textbf{A}}\eta = {\textbf{B}}$ has a unique solution.

Proof. It can be proved that ${\textbf{A}}$ is nonsingular. Let $\eta$ satisfy ${\textbf{A}}\eta = 0$, that is,

So, we can get the following equations:

By adding the above $(n+1)^2$ equations, we can get

So,

Note that $\mathcal{L}$ is reversible. Therefore, $\eta_{ij} = 0, \; \; i, j\in \mathbb{N}_n.$

According to Theorem 3.5, $u_n(t, x)$ can be obtained by substituting $\eta = A^{-1}B$ into $u_n = \sum\limits_{i = 0}^{n}\sum\limits_{j = 0}^{n}\eta_{ij}\rho_{ij}(t, x)$.

3.2.   Two-dimensional heat conduction equation

Similar to definition 2.2, we can give the definition of linear space $W_3[0, b]$ as follows:

Similarly, let $\{Q_n(y)\}_{n = 0}^\infty$ is the orthonormal basis in $L^2[0, b]$, and denote $Q_n(y) = \sum\limits_{k = 0}^n q_k y^k$.

Let

it is easy to prove that $\{JQ_n(y)\}_{n = 0}^\infty$ is the orthonormal basis of $W_3[0, b]$.

Let $\Omega = [0, 1]\times[0, a]\times[0, b]$. Now we define a three-dimensional space.

Definition 3.3 $W(\Omega) = \{u(t, x, y)|\frac{\partial^2 u}{\partial x \partial y}\in CC, (t, x, y)\in \Omega, u(0, x, y) = 0$, $u(t, 0, y) = u(t, a, y) = 0, u(t, x, 0) = u(t, x, b) = 0, \frac{\partial^5 u}{\partial t\partial x^2\partial y^2}\in L^2(\Omega)\},$ and

Similarly, we give the following theorem without proof.

Theorem 3.6. $\{\rho_{ijk}(t, x, y)\}_{i, j, k = 0}^\infty {\mathit{\mbox{is an orthonormal basis of}}}\; W(\Omega)$, where

Therefore, we can get $u_n$ as

according to the theory in Section 3.1, we can find all $\eta_{ijk}, \; \; i, j, k\in \mathbb{N}_n.$

3.3.   Three-dimensional heat conduction equation

By Lemma 2.1, note that the orthonormal basis of $L_2[0, c]$ is $\{R_n(z)\}_{n = 0}^\infty$, and denote $R_n(z) = \sum\limits_{m = 0}^n r_m z^m$, where $r_m$ is the coefficient of polynomial $R_n(z)$.

We can further obtain the orthonormal basis $JR_n(z) = \sum\limits_{m = 0}^n r_m \frac{z^{m+2}-c^{m+1}z}{(m+1)(m+2)}$ of $W_4[0, c]$, where

and

Let $G = [0, 1]\times[0, a]\times[0, b]\times[0, c]$. Now we define a four-dimensional space.

Definition 3.4. $W(G) = \{u(t, x, y, z)|\frac{\partial^3 u}{\partial x \partial y \partial z}\in CC, (t, x, y, z)\in G, u(0, x, y, z) = 0, u(t, 0, y, z) = u(t, a, y, z) = 0$, $u(t, x, 0, z) = u(t, x, b, z) = 0, u(t, x, y, 0) = u(t, x, y, c) = 0, \frac{\partial^7 u}{\partial t\partial x^2\partial y^2\partial z^2}\in L^2(G)\},$ and

where dG = dtdxdydz.

Similarly, we give the following theorem without proof.

Theorem 3.7. $\{\rho_{ijk}(t, x, y, z)\}_{i, j, k, m = 0}^\infty \mathit{\mbox{is an orthonormal basis of}}\; W(G)$, where

Therefore, we can get $u_n$ as

according to the theory in Section 3.1, we can find all $\eta_{ijkm}, \; \; i, j, k, m\in\mathbb{N}_n.$

4.   Convergence analysis

Suppose $u(t, x) = \sum\limits_{i = 0}^{\infty}\sum\limits_{j = 0}^{\infty}\eta_{ij}\rho_{ij}(t, x)$ is the exact solution of Eq (3.5). Let $P_{N_1, N_2}u(t, x) = \sum\limits_{i = 0}^{N_1} \sum\limits_{j = 0}^{N_2} \eta_{ij}T_i(t)P_j(x)$ is the projection of $u$ in $L(D)$.

Theorem 4.1. Suppose $\dfrac{\partial^{r+l} u(t, x)}{\partial t^{r}\partial x^{l}}\in L^2(D)$, and $N_1 > r, N_2 > l$, then, the error estimate of $P_{N_1, N_2}u(t, x)$ is

where $C$ is a constant, $N = min\{N_1, N_2\}, \alpha = min\{r, l\}.$

Proof. According to the lemma in ref. [29], it follows that

where $u_{N_1} = P_{t, N_1}u$ represents the projection of $u$ on variable $t$ in $L^2[0, 1]$, and $||\cdot||_{L_t^2[0, 1]}$ represents the norm of $(\cdot)$ with respect to variable $t$ in $L^2[0, 1]$.

By integrating both sides of the above formula with respect to $x$, we can get

Moreover,

According to the knowledge in Section 3,

where $c_{ij} = \langle \langle u, T_i\rangle _{L_t^2[0, 1]}, P_j\rangle_{L_x^2[0, a]}$.

Therefore,

Similarly,

In conclusion,

Theorem 4.2. Suppose $\dfrac{\partial^{r+l} u(t, x)}{\partial t^{r}\partial x^{l}}\in L^2(D)$, $u_n(t, x)$ is the $\varepsilon-$best approximate solution of Eq (3.2), and $n > max\{r, l\}$, then,

where $C$ is a constant, $\alpha = min\{r, l\}.$

Proof. According to Theorem 3.4 and Theorem 4.1, the following formula holds.

So, the $\varepsilon-$approximate solution has $\alpha$ convergence order, and the convergence rate is related to $n$, where represents the number of bases, and the convergence order can calculate as follows.

Where $n_i, i = 1, 2$ represents the number of orthonormal base elements.

5.   Numerical examples

Here, three examples are compared with other algorithms. $N$ represents the number of orthonormal base elements. For example, $N = 10 \times 10$, which means that we use the orthonormal system $\{\rho_{ij}\}_{i, j = 0}^{10}$ of $W(D)$ for approximate calculation, that is, we take the orthonormal system $\{JT_i(t)\}_{i = 0}^{10}$ and $\{JP_j(x)\}_{j = 0}^{10}$ to construct the $\varepsilon-$best approximate solution.

Example 5.1. Consider the following one-demensional heat conduction system ^[7,20]

The exact solution of Ex. 5.1 is $e^{-t}\sin x$.

In Table 1, $C.R.$ is calculated according to Eq (4.2). The errors in Tables 1 and 2 show that the proposed algorithm is very effective. In Figures 1 and 2, the blue surface represents the surface of the real solution, and the yellow surface represents the surface of $u_n$. With the increase of $N$, the errors between the two surfaces will be smaller.

Example 5.2. Consider the following two-demensional heat conduction system ^[20,21]

The exact solution of Ex. 5.2 is $u = e^{-2\pi^2 t}\sin(\pi x)\sin(\pi y)$.

Example 5.2 is a two-dimensional heat conduction equation. Table 3 shows the errors comparison with other algorithms. Table 4 lists the errors variation law in the $x-$axis direction. Figures 3 and 4 show the convergence effect of the scheme more vividly.

Example 5.3. Consider the three-demensional problem as following:

The exact solution of Ex. 5.3 is $u = e^{-\pi^2t}\sin(\dfrac{\pi x}{a})\sin(\dfrac{\pi y}{b})\sin(\dfrac{\pi z}{c})$.

Example 5.3 is a three-dimensional heat conduction equation, this kind of heat conduction system is also the most common case in the industrial field. Table 5 lists the approximation degree between the $\varepsilon-$best approximate solution and the real solution when the boundary time $t = 1$.

6.   Conclusions

The Shifted-Legendre orthonormal scheme is applied to high-dimensional heat conduction equations. The algorithm proposed in this paper has some advantages. On the one hand, the algorithm is evolved from the algorithm for solving one-dimensional heat conduction equation, which is easy to be understood and expanded. On the other hand, the standard orthogonal basis proposed in this paper is a polynomial structure, which has the characteristics of convergence order.

Acknowledgements

This work has been supported by three research projects (2019KTSCX217, 2020WQNCX097, ZH22017003200026PWC).

Conflict of interest

The authors declare no conflict of interest.