Generator functions and their applications

Abstract

We had introduced so called generators functions to precisely follow the regularity of analytic solutions of Navier-Stokes equations earlier (see Grenier and Nguyen [Ann. PDE 5 (2019)]. In this short note, we give a short presentation of these generator functions and use them to construct analytic solutions to classical evolution equations, which provides an alternative way to the use of the classical abstract Cauchy-Kovalevskaya theorem (see Asano [Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), pp. 102–105], Baouendi and Goulaouic [Comm. Partial Differential Equations 2 (1977), pp. 1151–1162], Caflisch [Bull. Amer. Math. Soc. (N.S.) 23 (1990), pp. 495–500], Nirenberg [J. Differential Geom. 6 (1972), pp. 561–576], Safonov [Comm. Pure Appl. Math. 48 (1995), pp. 629–637]).

1. Introduction

The general abstract Cauchy-Kovalevskaya theorem has been intensively used to construct analytic solutions to various evolution partial differential equations, including first order hyperbolic and parabolic equations, or Euler and Navier-Stokes equations. We refer for instance to Asano 1, Baouendi and Goulaouic 2, Caflisch 6, Nirenberg 16, Safonov 17, among others. In this short note, we use generators functions as introduced in 8 to obtain existence results for these equations. The results in this paper are not new, but show the versatility and simplicity of use of these generator functions. We believe that the approach could be used on many other equations and provide easy ways to obtain analytic solutions, including those at the large time, and to investigate instabilities.

Let us first introduce generator functions in the particular case of a periodic function $f(t,x)$ on $t \ge 0$ and $x\in \mathbb{T}^d$, $d\ge 1$. For $z \in {\mathbb{R}}$, we introduce the generator function $Gen[f]$ defined by

$$\begin{equation} \begin{aligned} Gen[f](t,z) &= \sum _{\alpha \in \mathbb{Z}^d} e^{z |\alpha |} |f_\alpha (t) | \end{aligned} {\tag{1.1}} \end{equation}$$

in which $f_\alpha$ denotes the Fourier transform of $f(t,x)$ with respect to $x\in \mathbb{T}^d$. If $f$ is analytic in $x$, $Gen[f]$ is only defined for small enough $|z|$, up to the analyticity radius of $f(t,.)$. The results in this note also apply to the case when $x\in \mathbb{R}^d$, with which the above summation is replaced by the integral over $\mathbb{R}^d$. In applications, we may also introduce generator functions depending on multi-variables $z = (z_1,\cdots ,z_d)$ that correspond to the analyticity radius of $f(x)$ in $(x_1,\cdots , x_d)$, respectively; see, for instance, 8 for the case of boundary layers on the half space $\mathbb{T}\times \mathbb{R}_+$.

First note that generator functions are non negative, and that all their derivatives are non negative and non decreasing in $z$. Moreover generator functions have very nice properties with respect to algebraic operations and differentiation. Namely they “commute” with the product, the sum and the differentiation, making their use very versatile.

Lemma 1.1.

For any $f,g$, there hold the following properties

$$\begin{equation} Gen [f + g] \le Gen[f] + Gen[g] {\tag{1.2}} \end{equation}$$

$$\begin{equation} Gen [fg] \le Gen[f] Gen[g] {\tag{1.3}} \end{equation}$$

$$\begin{equation} Gen[\nabla _xf] = \partial _zGen[f], \qquad \partial _t Gen[f] \le Gen[\partial _t f] ,\qquad \quad {\tag{1.4}} \end{equation}$$

for all $z\ge 0$.

Proof.

Let $f_\alpha$ and $g_\alpha$ be the Fourier transform of $f$ and $g$, respectively. It follows that

$$\begin{equation*} (fg)_\alpha = f_\alpha \star _\alpha g_\alpha \end{equation*}$$

for $\alpha \in \mathbb{Z}^d$. For $z\ge 0$, we compute

$$\begin{equation*} \begin{aligned} Gen[fg](z) &= \sum _{\alpha \in \mathbb{Z}^d} e^{z |\alpha |} |f_\alpha \star _\alpha g_\alpha | \le \sum _{\alpha \in \mathbb{Z}^d} \sum _{\beta \in \mathbb{Z}^d} e^{z |\beta |} e^{z|\alpha -\beta |} |f_{\beta }| |g_{\alpha -\beta } | \\ &\le Gen[f] Gen[g] \end{aligned} \end{equation*}$$

which is the second inequality. The first and third identities follow directly from the definition. The last inequality is a direct consequence of $\partial _t | f_\alpha | \le | \partial _t f_\alpha |.$

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The use of generator functions are not limited to polynomial operations. Namely, we have

Lemma 1.2.

Let $F, \widetilde{F}$ be analytic functions

$$\begin{equation*} F(z) = \sum _{n \ge 0} a_n z^n , \qquad \widetilde{F}(z) = \sum _{n \ge 0} | a_n | z^n. \end{equation*}$$

with some convergence radius $\rho$. Then, for any function $f(x)$, provided $\| f \|_{L^\infty } < \rho$, there holds

$$\begin{equation} Gen[F(f)] \le \widetilde{F} (Gen[f]) . \tag{1.5} \end{equation}$$

Proof.

First, using the algebra 1.3, we have

$$\begin{equation*} Gen[a_n f^n] \le |a_n| Gen[f]^n . \end{equation*}$$

Multiplying the inequality by $z^n$ and summing over $n$ give the lemma.

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2. An analyticity framework

In this section, we present our analyticity framework to construct analytic solutions via generator functions defined as in 1.1. The main point of the approach is that if $u(t,x)$ satisfies a non linear partial differential equation then $Gen[u](t,z)$ satisfies a transport differential inequality; see 2.7. This inequality describes in an acute way how the radius of analyticity shrinks as time goes on, and allows to get analytic bounds on $u$, and in particular to bound all its derivatives at the same time.

To precise the framework, we consider the following general system of evolution equations

$$\begin{equation} \partial _t u = A(u, \nabla _x u) {\tag{2.1}} \end{equation}$$

for a vector valued function $u = u(x,t)$, with $x\in \mathbb{T}^d$, $d\ge 1$, and $t\ge 0$. We assume that the function $A(\cdot ,\cdot )$ satisfies

$$\begin{equation} Gen[A(u,\nabla _x u)] \le C_0 F( Gen[u] ) \Big ( 1 + \partial _z Gen[u]\Big ) {\tag{2.2}} \end{equation}$$

for some constant $C_0$ and some analytic functions $F$. For instance, $A$ may be a quadratic polynomial in $u$ and $\nabla u$, in which case $F = Id$, or more generally, $A$ may be of the form $G(u) \cdot \nabla _x u$, for which $F = \widetilde{G}$ (see Lemma 1.2). Note that we do not make any assumption on the hyperbolicity of the system 2.1.

We shall construct solutions in the function space $X_\rho$ defined by

$$\begin{equation} X_\rho : = \Big \{ u\nobreakspace :\nobreakspace Gen[u](\rho ) <\infty \Big \}. {\tag{2.3}} \end{equation}$$

Theorem 2.1.

Let $\rho >0$ and $u_0$ be in $X_{\rho }$. Then the Cauchy problem 2.1 with initial data $u_0$ has a unique solution $u(t)$ in $X_{\rho (t)}$ for positive times $t$ as long as $\rho (t) = \rho - C_1 t$ remains positive, $C_1$ being some large positive constant depending on $u_0$.

Proof.

We shall construct solutions via the standard approximation. First, we let $P_N$ be the projection of the first $N$ Fourier modes; namely

$$\begin{equation} P_N(f)(x) = \sum _{\alpha \in \mathbb{Z}^d, |\alpha | \le N} e^{i \alpha x} f_\alpha .{\tag{2.4}} \end{equation}$$

Let $u_N(t)$ be a solution to the following regularized equations

$$\begin{equation} \partial _t u_N = P_N A(P_N(u_N), \nabla _x P_N (u_N)) \tag{2.5} \end{equation}$$

with initial data $u_N(0,x) = P_N u_0(x)$ for all $N$. As the right hand side consists of only a finite number of Fourier modes, this equation is in fact an ordinary differential equation. Hence there exists a unique solution $u_N(t)$, defined for $t$ small enough. It suffices to prove that $u_N(t)$ is a Cauchy sequence in $X_{\rho (t)}$, as long as $\rho (t)$ remains positive.

The Fourier coefficients of the solution to 2.1 satisfy

$$\begin{equation*} \partial _t u_{N,\alpha } =P_{N,\alpha } A(P_N u_N, \nabla _x P_N u_N) \end{equation*}$$

where $P_{N,\alpha } f$ denotes the Fourier coefficient $f_\alpha$ if $|\alpha |\le N$, and zero, if otherwise. We therefore get

$$\begin{equation*} \begin{aligned} \sum _{| \alpha | \le N} \partial _t | u_{N,\alpha } | e^{z |\alpha |} &\le \sum _{| \alpha | \le N} | \partial _t u_{N,\alpha } | e^{z |\alpha |} \\ & \le \sum _{| \alpha | \le N} | P_{N,\alpha } A(P_N u_N, \nabla _x P_N u_N) | e^{z |\alpha |}. \end{aligned} \end{equation*}$$

By definition, we note that

$$\begin{equation} Gen[P_N(f)] \le Gen[f] . {\tag{2.6}} \end{equation}$$

Thus, using the assumption 2.2, we obtain the following Hopf-type differential inequality

$$\begin{equation} \partial _t Gen[u_N]\le C_0 F(Gen[u_N]) \Big ( 1 + \partial _z Gen[u_N]\Big ), {\tag{2.7}} \end{equation}$$

in which $C_0$ is independent of $N$. For convenience, we set

$$\begin{equation*} G_N(t,z) = Gen[u_N(t)](z). \end{equation*}$$

The previous inequality yields

$$\begin{equation} \partial _t G_N \le C_0 F(G_N) (1+ \partial _z G_N), {\tag{2.8}} \end{equation}$$

which is a differential inequality that we will now exploit in order to bound $u_N$ and all its derivatives. Note that $G_N$ is a finite sum and is therefore defined for all $z$. As $N$ goes to $+ \infty$, $G_N(0,z)$ converges to $Gen[u_0](z)$, which is defined for $|z| \le \rho$.

As usual with analytic solutions, the domain of analyticity shrinks with time, hence we introduce

$$\begin{equation*} F_N(t,z) = G_N(t,\theta (t) z) \end{equation*}$$

for $t, z\ge 0$, where $\theta (\cdot )$ will be determined later. It follows that $F_N$ satisfies

$$\begin{equation} \partial _t F_N \le C_0 F(F_N) + C_0(F( F_N) + \theta '(t) z) \partial _z F_N. {\tag{2.9}} \end{equation}$$

Note that $F_N$ is defined for any $z$. In the limit $N \to + \infty$ we focus on $0 \le z \le \rho$.

We will choose $\theta (t)$ in such a way that the characteristics of 2.9 are outgoing on $[0,\rho ]$, namely such that at $z =0$, $F(F_N) > 0$ (which is always satisfied) and such that at $z =\rho$, $F(F_N) + \theta '(t) \rho < 0$. At $t= 0$, we choose $\theta (0) =1$ and thus

$$\begin{equation*} F_N(0,z) = G_N(0,z) = Gen[P_N u_0](z)\le Gen[u_0](z), \end{equation*}$$

which is well-defined on $[0,\rho ]$. Set

$$\begin{equation*} M_0 = \sup _{0 \le z \le \rho } Gen[u_0](z). \end{equation*}$$

We will focus on times $0 \le t \le T_N$ such that $F_N(t,z) \le 2 M_0$ for $0 \le t \le T_N$ and $0 \le z \le \rho$. We choose

$$\begin{equation*} \theta (t) = 1 - F(3 M_0) \rho ^{-1} t . \end{equation*}$$

Observe that on $0 \le t \le T_N$, as long as $\theta (t) > 0$, $F(F_N) + \theta '(t) \rho <0$. On such a time interval, 2.9 is a nonlinear transport equation with a source term and with outgoing characteristics at $0$ and $\rho$. As a consequence, we have

$$\begin{equation*} \partial _t \sup _{0 \le z \le 1} F_N(t,z) \le C_0 \sup _{0 \le z \le 1} F(F_N(t,z)) \le C_0 F(2 M_0), \end{equation*}$$

hence

$$\begin{equation*} F_N(t,z) \le M_0 + C_0 t F(2 M_0). \end{equation*}$$

Classical arguments then lead to the fact that $T_N$ is bounded away from $0$, namely there exists some $T > 0$ such that $T_N \ge T$ for any $N$, and such that $\rho (t) > 0$ for any $t \le T$. This implies that

$$\begin{equation*} F_N(t,z) \le C(F_N(0,z)) \le C(M_0) \end{equation*}$$

for any $0 \le z \le \rho$ and any $0 \le t \le T$. Thus $u_N(t)$ is uniformly bounded in $X_{\rho (t)}$ for all $N$. As a consequence, $u_N$ and all its derivatives of all orders are uniformly bounded in $L^\infty$. Up to the extraction of a subsequence, $u_N$ and all its derivatives converge uniformly, towards some function $u$, solution of 2.1. Moreover, classical arguments show that $u(t) \in X_{\rho (t)}$ for any $0 \le t \le T$, which ends the proof of the theorem.

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3. Euler equations

In this section, we apply the previous framework to construct analytic solutions to incompressible Euler equations on $\mathbb{T}^d$ or $\mathbb{R}^d$, $d\ge 2$. Namely, we consider

$$\begin{equation} \begin{aligned} \partial _t u + u \cdot \nabla u + \nabla p &=0 \\ \nabla \cdot u &=0 \end{aligned} {\tag{3.1}} \end{equation}$$

on $\mathbb{T}^d$ (the case $\mathbb{R}^d$ is treated similarly). Again, the existence result is classical (see, for instance, 341213). The function space $X_\rho$ is defined in 2.3. We have

Theorem 3.1.

Let $\rho >0$ and $u_0$ be a divergence-free vector field in $X_{\rho }$. Then the Cauchy problem 3.1 with initial data $u_0$ has a solution $u(t)$ in $X_{\rho (t)}$ for positive times $t$ as long as $\rho (t) = \rho - C_1 t$ remains positive, $C_1$ being a constant depending on $u_0$.

Proof.

Introduce the Leray projector $\mathbb{P}$, projection onto the divergence-free $L^2$ vector space. In Fourier coefficients, $\mathbb{P}_\alpha$ is an $d\times d$ matrix with entries

$$\begin{equation*} (\mathbb{P}_\alpha )_{jk} = \delta _{jk} - \frac{\alpha _j\alpha _k}{|\alpha |^2}. \end{equation*}$$

In particular, $|\mathbb{P}_\alpha |$ is bounded, uniformly in $\alpha \in \mathbb{Z}^d$. Taking the Leray projection of 3.1, we obtain

$$\begin{equation} \partial _t u = - \mathbb{P} (u\cdot \nabla u), {\tag{3.2}} \end{equation}$$

which falls into the previous abstract framework. It remains to check the assumption 2.2. Indeed, using 1.2 and the uniform boundedness of $|\mathbb{P}_\alpha |$ in $\alpha$, we compute

$$\begin{equation*} Gen[\mathbb{P} (u\cdot \nabla u)] \le Gen[u\cdot \nabla u] \le Gen[u] Gen[\nabla u] \le Gen[u]\partial _z Gen[u]. \end{equation*}$$

Note that the divergence-free condition is invariant under 3.2. Thus, applying the abstract framework introduced in the previous section to the evolution equation 3.2, we obtain Theorem 3.1.

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Remark 3.2.

Note that there is no hyperbolicity assumption made in the first order evolution equation 2.1, which may in particular be illposed in Sobolev spaces. The abstract framework can also be applied to a variety of other physical relevant models (e.g., 711) that arise in a singular limit of Euler equations, Navier-Stokes equations, and Vlasov-Poisson systems.

4. Comments on other applications

In this section, we briefly highlight two recent applications to the use of generator functions to capture the instability of boundary layers 8 and prove the nonlinear Landau damping 910, both of which have a different flavor from the previous short time existence theory. These applications 8910 are a version of the large time Cauchy-Kovalevskaya theorem. The use of generator functions allows us to control all the derivatives uniformly in the small viscosity limit or in the large time limit. In particular, the work 9 provides an elementary proof of the nonlinear Landau damping that was first obtained by Mouhot and Villani 15 for analytic data and by Bedrossian, Masmoudi, and Mouhot 5 for Gevrey data.