A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids

1.   Introduction

Let $(M, \omega)$ be a complex $n$-dimensional compact Hermitian manifold and $\chi$ be a smooth real (1, 1)-form on $(M, \omega)$. $\Gamma_k^{\omega}$ is the set of all real (1, 1)-forms whose eigenvalues belong to the $k$-positive cone $\Gamma_k$. For any $u\in C^2(M)$, we can get a new (1, 1)-form

In any local coordinate chart, $\chi_u$ can be expressed as

In this article, we study the following form of parabolic Hessian quotient equations

where $0\leq l < k \leq n$, $[0, T)$ is the maximum time interval in which the solution exists and $\phi(x, z)\in C^{\infty}(M\times \mathbb{R})$ is a given strictly positive function.

The study of the parabolic flows is motivated by complex equations

Equation (1.2) include some important geometry equations, for example, complex Monge-Ampère equation and Donaldson equation ^[6], which have attracted extensive attention in mathematics and physics since Yau's breakthrough in the Calabi conjecture ^[28]. Since Eq (1.2) are fully nonlinear elliptic, a classical way to solve them is the continuity method. Using this method, the complex Monge-Ampère equation

was solved by Yau ^[28]. Donaldson equation

was independently solved by Li-Shi-Yao ^[11], Collins-Szèkelyhidi ^[3] and Sun ^[17]. Equation (1.2) also include the complex $k$-Hessian equation and complex Hessian quotient equation, which, respectively, correspond to

Dinew and Kolodziej ^[7] proved a Liouville type theorem for $m$-subharmonic functions in $\mathbb{C}^n$, and combining with the estimate of Hou-Ma-Wu ^[10], solved the complex $k$-Hessian equation by using the continuity method. Under the cone condition, Sun ^[16] solved the complex Hessian quotient equation by using the continuity method. There have been many extensive studies for complex Monge-Ampère equation, Donaldson equation, the complex $k$-Hessian equation and the complex Hessian quotient equation on closed complex manifolds, see, e.g., ^{[4,12,20,22,23,29,30]}. When the right hand side function $\phi$ in Eq (1.2) depends on $u$, that is $\phi = \phi(x, u)$, it is interesting to ask whether we can solve them. We intend to solve (1.2) by the parabolic flow method.

Equation (1.1) covers some of the important geometric flows in complex geometry. If $k = n$ and $l = 0$, (1.1) is known as the complex Monge-Ampère flow

which is equivalent to the Kähler-Ricci flow. The result of Yau ^[28] was reproduced by Cao ^[2] through Kähler-Ricci flow. Using the complex Monge-Ampère flow, similar results on a compact Hermitian manifold and a compact almost Hermitian manifold were proved by Gill ^[9] and Chu ^[5], respectively. To study the normalized twisted Chern-Ricci flow

which is equivalent to the following Mong-Ampère flow

Tô ^[25,26] considered the following complex Monge-Ampère flow

where $\Omega$ is a smooth volume form on $M$. From this, we can see that the given function $\phi$ depends on $u$ in some geometric flows. If $l = 0$, (1.1) is called as the complex $k$-Hessian flow

The solvability of complex $k$-Hessian flow was showed by Sheng-Wang ^[21].

In this paper, our research can be viewed as a generalization of Tô's work in ^[26] and Sheng-Wang's work in ^[21]. To solve the complex Hessian quotient flow, the condition of the parabolic $\mathcal{C}$-subsolution is needed. According to Phong and Tô ^[14], we can give the definition of the parabolic $\mathcal{C}$-subsolution to Eq (1.1).

Definition 1.1. Let $\underline{u}(x, t)\in C^{2, 1}(M\times [0, T))$ and $\chi_{\underline{u}}\in\Gamma_k^{\omega}$, if there exist constants $\delta$, $R > 0$, such that for any $(x, t)\in M\times [0, T)$,

implies that

then $\underline{u}$ is said to be a parabolic $\mathcal{C}$-subsolution of (1.1), where $\lambda(\underline{u})$ denotes eigenvalue set of $\chi_{\underline{u}}$.

Obviously, we can give the equivalent definition of parabolic $\mathcal{C}$-subsolution of (1.1).

Definition 1.2. Let $\underline{u}(x, t)\in C^{2, 1}(M\times [0, T))$ and $\chi_{\underline{u}}\in\Gamma_k^{\omega}$, if there exist constant $\widetilde{\delta} > 0$, for any $(x, t)\in M\times [0, T)$, such that

then $\underline{u}$ is said to be a parabolic $\mathcal{C}$-subsolution of (1.1).

Our main result is

Theorem 1.3. Let $(M, g)$ a compact Hermitian manifold and $\chi$ be a smooth real $(1, 1)$-form on $M$. Assume there exists a parabolic $\mathcal{C}$-subsolution$\underline{u}$ for Eq $(1.1)$ and

where $c_{\phi}$ is a constant.Then there exits a unique smooth solution $u(x, t)$ to $(1.1)$ all timewith

Moreover, $u(x, t)$ is$C^\infty$ convergent to a smooth function $u_\infty$, which solves Eq $(1.2)$.

The rest of this paper is organized as follows. In Section 2, we give some important lemmas and estimate on $|u_t(x, t)|$. In Section 3, we prove $C^0$ estimates of Eq (1.1) by the parabolic $\mathcal{C}$-subsolution condition and the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, using the parabolic $\mathcal{C}$-subsolution condition, we establish the $C^2$ estimate for Eq (1.1) by the method of Hou-Ma-Wu ^[10]. In Section 5, we adapt the blowup method of Dinew and Kolodziej ^[7] to obtain the gradient estimate. In Section 6, we give the proof of the long-time existence of the solution to the parabolic equation and its convergence, that is Theorem 1.3.

2.   Preliminaries

In this section, we give some notations and lemmas. In holomorphic coordinates, we can set

$\lambda(u)$ and $\lambda(\underline{u})$ denote the eigenvalue set of $\{X_{i\overline{j}}\}$ and $\{\underline{X}_{i\overline{j}}\}$ with respect to $\{g_{i\overline{j}}\}$, respectively. In local coordinates, (1.1) can be written as

For simplicity, set

then (2.1) is abbreviated as

We use the following notation

For any $x_0\in M$, we can choose a local holomorphic coordinates such that the matrix $\{X_{i\overline{j}}\}$ is diagonal and $X_{1\overline{1}}\geq\cdots\geq X_{n\overline{n}}$, then we have, at $x_0\in M$,

To prove a priori $C^0$-estimate for solution to Eq (1.1), we need the following variant of the Alexandroff-Bakelman-Pucci maximum principle, which is Proposition 10 in ^[20].

Lemma 2.1. ^[20] Let $v:B(1)\rightarrow \mathbb{R}$ be a smooth function, which meets the condition$v(0)+\epsilon\leq \inf_{\partial B(1)}v$, where$B(1)$ denotes the unit ball in $\mathbb{R}^n$.Define the set

Then there exists a costant $c_0 > 0$ such that

Next, we give an estimate on $|u_t(x, t)|$.

Lemma 2.2. Under the assumption of Theorem 1.3, let $u(x, t)$ be a solution to $(1.1)$. Then for any $(x, t)\in M\times [0, T)$, we have

Furthermore, there is a constant $C > 0$ such that

where $C$ depends on $H = |u_0|_{C^2(M)}$ and $|\phi|_{C^0(M\times [-H, H])}$.

Proof. Differentiating (2.2) on both sides simultaneously at $t$, we obtain

Set $u_t^{\varepsilon} = u_t-\varepsilon t, \ \varepsilon > 0$. For any $T'\in (0, T)$, suppose $u_t^{\varepsilon}$ achieves its maximum $M_t$ at $(x_0, t_0)\in M\times [0, T']$. Without loss of generality, we may suppose $M_t\geq 0$. If $t_0 > 0$, From the parabolic maximum principle and (2.4), we get

This is obviously a contradiction, so $t_0 = 0$ and

that is

Letting $\varepsilon\rightarrow 0$, we obtain

Since $T'\in (0, T)$ is arbitrary, we have

Similarly, setting $u_t^{\varepsilon} = u_t+\varepsilon t, \ \varepsilon > 0$, we obtain

(2.1) yields

Combining (2.5)–(2.7), we complete the proof of Proposition 2.2.

From the concavity of $F(\lambda(u))$ and the condition of the parabolic $\mathcal{C}$-subsolution, we give the following lemma, which plays an important role in the estimation of $C^2$.

Lemma 2.3. Under the assumption of Theorem 1.3 and assuming that $X_{1\overline{1}}\geq\cdots\geq X_{n\overline{n}}$, there exists two positive constants$N$ and $\theta$ such that we have either

or

Proof. Since $\underline{u}$ is a parabolic $\mathcal{C}$-subsolution to Eq (1.1), from Definition 1.2, there are uniform constants $\widetilde{\delta} > 0$ and $N > 0$, such that

If $\epsilon > 0$ is sufficiently small, it can be obtained from (2.10)

Set $\lambda' = \lambda(\underline{u})-\epsilon I+Ne_1$, then

Using the concavity of $F(\lambda(u))$ gives

From Lemma 2.2 and (1.4), we obtain

In addition, it can be obtained from the condition (1.6)

(2.13) and (2.14) deduce that

It follows from this that

Combining (2.2), (2.11) and (2.16) gives that

Put (2.17) into (2.12)

Let

If $F^{1\overline{1}}N\leq \theta(1+\mathcal{F})$, Inequality (2.8) is obtained, otherwise Inequality (2.9) must be true.

3.   $C^0$ Estimates

In this section, we prove the $C^0$ estimates by the existence of the parabolic $\mathcal{C}$-subsolution and the Alexandroff-Bakelman-Pucci maximum principle.

Proposition 3.1. Under the assumption of Theorem1.3, let $u(x, t)$ be a solution to (1.1). Then there exists a constant $C > 0$ such that

where $C$ depends on $|u_0|_{C^2(M)}$ and $|\underline{u}|_{C^2(M\times [0, T))}$.

Proof. Combining (2.13), (2.14) and $\frac{\partial\phi(x, z)}{\partial z}\geq0$ yields

Let's rewrite Eq (2.2) as

when fix $t\in[0, T)$, Eq (3.2) is elliptic. From (3.1), we see that the parabolic $\mathcal{C}$-subsolution $\underline{u}(x, t)$ is a $\mathcal{C}$-subsolution to Eq (3.2) in the elliptic sense. From (2.15), we have

Our goal is to obtain a lower bound for $L = \inf_{M\times t}(u-\underline{u})$. Note that $\lambda(u)\in \Gamma_{k}$, which implies that $\lambda(u)\in \Gamma_{1}$, then $\Delta(u-\underline{u})\geq -\widetilde{C}$, where $\Delta$ is the complex Laplacian with respect to $\omega$. According to Tosatti-Weinkove's method ^[22], we can prove that $\|u-\underline{u}\|_{L^1(M)}$ is bounded uniformly. Let $G:M\times M\rightarrow \mathbb{R}$ be the associated Green's function, then, by Yau ^[28], there is a uniform constant $K$ such that

Since

then for fixed $t\in [0, T)$ there exists a point $x_0\in M$ such that $(u-\underline{u})(x_0, t) = 0$. Thus

that is

Let us work in local coordinates, for which the infimum $L$ is achieved at the origin, that is $L = u(0, t)-\underline{u}(0, t)$. We write $B(1) = \{z:|z| < 1\}$. Let $v = u-\underline{u}+\epsilon|z|^2$, for a small $\epsilon > 0$. We have $\inf v = L = v(0)$, and $v(z)\geq L+\epsilon$ for $z\in\partial B(1)$. From Lemma 2.1, we get

At the same time, if $x\in \Omega$, then $D^2v(x)\geq 0$ implies that

If $\epsilon$ is sufficiently small, then

Set $\mu = \lambda(u)-\lambda(\underline{u})$. Since $\lambda(u)$ satisfies Eq (3.2), then

$\underline{u}$ is a $\mathcal{C}$-subsolution to Eq (3.2) in the elliptic sense, so there is a uniform constant $R > 0$, such that

which means $|v_{i\overline{j}}|\leq C,$ for any $x\in \Omega$. As in Blocki ^[1], for $x\in \Omega$, we have $D^2v(x)\geq 0$ and so

From this and (3.3), we obtain

On the other hand, by the definition of $\Omega$ in Lemma 2.1, for $x\in \Omega$, we get

and so

It follows that

Since $\|u-\underline{u}\|_{L^1(M)}$ is bounded uniformly, $\int_{M}|v(x)|$ is also bounded uniformly. If $L$ is very large, Inequality (3.6) contradicts (3.5), which means that $L$ has a lower bound. For any $t\in [0, T)$, Inequality (3.1) holds, thus

4.   $C^2$ Estimates

In this section, we prove that the second-order estimates are controlled by the square of the gradient estimate linearly. Our calculation is a parabolic version of that in Hou-Ma-Wu ^[10].

Proposition 4.1. Under the assumption of Theorem1.3, let $u(x, t)$ be a solution to (1.1). Then there exists a constant $\widetilde{C}$ such that

where $\widetilde{C}$ depends $\chi$, $\omega$, $|\phi|_{C^2(M\times [-C, C])}$, $|\underline{u}|_{C^2(M\times [0, T))}$, $|\partial_t\underline{u}|_{C^0(M\times [0, T))}$ and $|u_0|_{C^2(M)}$.

Proof. Let $\lambda(u) = (\lambda_1, \ldots, \lambda_n)$ and $\lambda_1$ is the maximum eigenvalue. For any $T' < T$, we consider the following function

where $\varphi$ and $\psi$ are determined later. We want to apply the maximum principle to the function $W$. Since the eigenvalues of the matrix $\{X_{i\overline{j}}\}$ with respect to $\omega$ need not be distinct at the point where $W$ achieves its maximum, we will perturb $\{X_{i\overline{j}}\}$ following the technique of ^[20]. Let $W$ achieve its maximum at $(x_0, t_0)\in M\times [0, T']$. Near $(x_0, t_0)$, we can choose local coordinates such that $\{X_{i\overline{j}}\}$ is diagonal with $X_{1\overline{1}}\geq\cdots\geq X_{n\overline{n}}$, and $\lambda(u) = (X_{1\overline{1}}, \cdots, X_{n\overline{n}})$. Let $D$ be a diagonal matrix such that $D^{11} = 0$ and $0 < D^{22} < \cdots < D^{nn}$ are small, satisfying $D^{nn} < 2D^{22}$. Define the matrix $\widetilde{X} = X-D$. At $(x_0, t_0)$, $\widetilde{X}$ has eigenvalues

Since all the eigenvalues of $\widetilde{X}$ are distinct, we can define near $(x_0, t_0)$ the following smooth function

where

and $C_1 > 0$ is to be chosen later. Direct calculation yields

and

Without loss of generality, we can assume that $\lambda_1 > 1$. From here on, all calculations are done at $(x_0, t_0)$. From the maximum principle, calculating the first and second derivatives of the function $\widetilde{W}$ gives

Define

Obviously,

Next, we will estimate the terms in (4.8). Direct calculation shows that

According to Inequality (78) in ^[20], we have

where $C_0$ depending $\chi$, $\omega$, $|\phi|_{C^2(M\times [-C, C])}$, $|\underline{u}|_{C^2(M\times [0, T))}$, $|\partial_t\underline{u}|_{C^0(M\times [0, T))}$ and $|u_0|_{C^2(M)}$). From here on, $C_0$ can always absorb the constant it represents before, and can change from one line to the next, but it does not depend on the parameter we choose later. By calculating the covariant derivatives of (4.7) in the direction $\frac{\partial}{\partial z^1}$ and $\frac{\partial}{\partial \overline{z}^1}$, we obtain

and

Notice that

therefore

Combining (4.14) with

gives

Let's set $\lambda_1\geq K$, otherwise the proof is completed. Putting 4.10–4.12 and (4.15) into (4.9) yields

A simple computation gives

Next, we estimate the formula (4.17). Differentiating Eq (2.2), we have

and

It follows from (4.18) and (4.19) that

By commuting derivatives, we have the identity

Direct calculation gives

It follows from (4.21) that

Noticing that

from this and (4.23), we obtain

In the same way, we can get

Using (4.20)–(4.26) in (4.17), we have

A simple calculation gives

Substituting (4.27), (4.28) and (4.16) into (4.8),

Let $\delta > 0$ be a sufficiently small constant to be chosen later and satisfy

We separate the rest of the calculations into two cases.

Case 1: $\lambda_{n} < -\delta \lambda_{1}$.

Using(4.5), we find that

From (4.13), we have

Combining (4.31) with (4.32), we conclude that

Note that the operator $F$ is concave, which implies that

Applying (4.33) and (4.34) to (4.29) and using $\varphi'' = 2(\varphi')^2$ yield that

Note that the fact

It follows that

Using the following inequality

deduces

Note that $\lambda_1 > 1$, we have

Since $\psi^{''} > 0$, which implies that

According to Lemma 2.3, there are at most two possibilities:

(1) If (2.8) holds true, then

Substituting (4.37)—(4.40) into (4.35) and using $\varphi'\geq \frac{1}{4K}$ yield that

We may set $\theta C_1\geq C_0$. It follows from (4.41) that $\lambda_1\leq \widetilde{C}K$.

(2) If (2.9) holds true,

According to $\psi^{'} < 0$ and the concavity of the operator $F$, we have

Using (4.37)—(4.39) and (4.43) in (4.35), together with (4.42), we find that

Let $\lambda_1$ be sufficiently large, so that

It follows from (4.44)that $\lambda_1\leq \widetilde{C}K$.

Case 2: $\lambda_{n}\geq-\delta \lambda_{1}$.

Let

Let us first treat those indices which are not in $I$. Similar to (4.31), we obtain

Using (4.32) yields that

Since

which implies that

It follows that

Recalling $\varphi'' = 2(\varphi')^2$ and $0 < \delta\leq\frac{1}{2}$, we obtain from (4.5) that

Notice that $\psi^{''}\geq\frac{4\epsilon}{1-\epsilon}(\psi^{'})^2$ if $\epsilon = \frac{1}{4E+1}$. Since $\frac{1}{4E+1}\geq \delta$, we get that

Take (4.46)–(4.49) into (4.29),

Similar to (4.37) and (4.38), by using the third term of (4.50) to absorb the first two terms of it, We get that

According to Lemma2.3, there are at most two possibilities:

(1) If (2.8) holds true, then

Put (4.52) into (4.51)

Here, $C_1$ is determined finally, such that

It follows from (4.53) that

(2) If (2.9) holds true,

Substituting (4.43) into (4.51) and using (4.54) give that

It follows that

5.   $C^1$ Estimates

To obtain the gradient estimates, we adapt the blow-up method of Dinew and Kolodziej ^[7] and reduce the problem to a Liouville type theorem which is proved in ^[7].

Proposition 5.1. Under the assumption of Theorem 1.3, let $u(x, t)$ be a solution to (1.1). Then there exists a uniform constant $\widetilde{C}$ such that

Proof. Suppose that the gradient estimate (5.1) does not hold. Then there exists a sequence $(x_m, t_m)\in M\times [0, T)$ with $t_m\rightarrow T$ such that

After passing to a subsequence, we may assume that $\lim_{m\rightarrow \infty}x_m = x_0\in M$. We choose a coordinate chart $\{U, (z_1, \cdots, z_n)\}$ at $x_0$, which we identify with an open set in $\mathbb{C}^n$, and such that $\omega(0) = \beta = \sqrt{-1}\sum_{i}dz^i\wedge d\overline{z}^i$. We may assume that the open set contains $\overline{B_1(0)}$ and $m$ is sufficiently large so that $z_m = z(x_m)\in B_1(0)$. Define

From this and Proposition 4.1, we have

This yields that $\widetilde{u}_{m}$ is contained in the Hölder space $C^{1, \gamma(\mathbb{C}^n)}$ with a uniform. Along with a standard application of Azela-Ascoli theorem, we may suppose $\widetilde{u}_{m}$ has a limit $\widetilde{u}\in C^{1, \gamma}(\mathbb{C}^n)$ with

in particular $\widetilde{u}$ is not constant. On the other hand, similar to the method of Dinew and Kolodziej ^[7], we have

Fixing $z$, we obtain

Lemma 2.2 gives $\partial_t u$ is bounded uniformly. Since

which implies that $\phi_m(\frac{z}{C_m}, u_m)$ is bounded uniformly. Taking the limits on both sides of 5.3 by $m\rightarrow \infty$ yields that

which is in the pluripotential sense. Moreover, a similar reasoning tells us that for any $1\leq p\leq k$,

Then, (5.4) and (5.5) imply $\widetilde{u}$ is a $k$-subharmonic. By a result of Blocki ^[1], $\widetilde{u}$ is a maximal $k$-subharmonic function in $\mathbb{C}^n$. Applying the Liouville theorem in ^[7], we find that $\widetilde{u}$ is a constant, which contradicts with (5.2).

6.   Long-time existence and convergence

In this section, we shall give a proof of the long-time existence to the flow and its convergence, that is Theorem 1.3.

From Lemma 2.2, Proposition 3.1, Propositions 4.1 and 5.1, we conclude that Eq 1.1 is uniformly parabolic. Therefore by Evans-Krylov's regularity theory ^[8,13,19,27] for uniformly parabolic equation, we obtain higher order derivative estimates. By the a priori estimates which don't depend on time, one can prove that the short time existence on $[0, T)$ extends to $[0, \infty)$, that is the smooth solution exists at all time $t > 0$. After proving $C^{\infty}$ estimates on $[0, \infty)$, we are able to show the convergence of the solution flow.

Let $v = e^{\gamma t}u_t$, where $0 < \gamma < c_{\phi}$. Commuting derivative of $v$ with respect to $t$ and using (2.4), we obtain

Using the condition (1.5) yields $\gamma-\frac{\phi_z}{\phi} < 0$, According to the parabolic maximum principle, it follows that

which means that $|u_t|$ decreases exponentially, in particular

According to Proposition 3.1, it follows that $u+\frac{C}{\gamma}e^{-\gamma t}$ is bounded uniformly and decreasing in $t$. Thus it converges to a smooth function $u_{\infty}$. From the higher order prior estimates, we can see that the function $u(x, t)$ converges smoothly to $u_{\infty}$. Letting $t\rightarrow \infty$ in Eq (2.1),

7.   Conclusions

In this paper, we have considered the parabolic Hessian quotient equation (1.1), in which the right hand side function $\phi$ depends on $u$. Firstly, we prove $C^0$ estimates of Eq (1.1) by the parabolic $\mathcal{C}$-subsolution condition and the Alexandroff-Bakelman-Pucci maximum principle. Secondly, we establish the $C^2$ estimate for Eq (1.1) by using the parabolic $\mathcal{C}$-subsolution condition. Thirdly, we obtain the gradient estimate by adapting the blowup method. Finally we give the proof of the long-time existence of the solution to the parabolic equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.

Acknowledgments

This work was supported by the Natural Science Foundation of Anhui Province Education Department (Nos. KJ2021A0659, gxgnfx2018017); Quality Enginering Project of Anhui Province Education Department (Nos. 2018jyxm0491, 2019mooc205, 2020szsfkc0686); Science Research Project of Fuyang Normal University (No. 2021KYQD0011)

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.