Generator functions and their applications

The Cauchy-Kovalevskaya theorem is a classical convenient tool to construct analytic solutions to partial differential equations, which allows one to view and treat them as if they are ordinary differential equations:

$\displaystyle u_t = F(t,x,u,u_x)$

for unknown functions ${u(t,x)}$ in ${t\ge 0}$ and ${x\in \mathbb{R}^d}$ (or some spatial domain). Roughly, if ${F(t,x,u,w)}$ is locally analytic near a point ${(0,x_0,u_0,w_0)}$ then the PDE has a unique solution ${u(t,x)}$ which is analytic near ${(0,x_0)}$ , as established by Cauchy (1842) and generalized by Kovalevskaya as part of her dissertation (1875). The theorem has found many applications such as in fluid dynamics and kinetic theory where it is used to provide existence of analytic solutions including those that are obtained at certain asymptotic limits (e.g., inviscid limit, Boltzmann-to-fluid limit, Landau damping, inviscid damping,…). There are modern formulations of the abstract theorem: see, for instance, Asano, Baouendi and Goulaouic, Caflisch, Nirenberg, and Safonov. In this blog post, I present generator functions, as an alternative approach to the use of the Cauchy-Kovalevskaya theorem, recently introduced in my joint work with E. Grenier (ENS Lyon), and discuss the versatility and simplicity of their use to applications. In addition to providing existence of analytic solutions, I will also mention another use of generator functions to capture some physics that would be otherwise missed for analytic data (namely, an analyticity framework to capture physical phenomena that are not seen for analytic data!).

1. Generator functions

By definition, analyticity of a function is to quantify all successive derivatives of the function in a controllable way. For instance, ${u(x)}$ is analytic near ${x = x_0}$ if and only if the infinite power series

$\displaystyle \sum_{n \ge 0} \partial_x^n u(x_0) \frac{ (x-x_0)^n }{n!}$

converges in a neighborhood of ${x}$ near ${x_0}$ . Analyticity radius ${z}$ is defined so that the series converges for ${x}$ in the ball of radius ${z}$ and centered at ${x_0}$ . This leads us to introduce generator functions

$\displaystyle Gen[u](z) = \sum_{n \ge 0} \| \partial_x^n u \| {z^n \over n !}$

defined for analyticity radius ${z\ge 0}$ and for analytic function ${u = u(x)}$ , where ${\|\cdot \|}$ is some convenient norm (say, ${L^\infty}$ norm). Namely, the generator function measures the analyticity of ${u(x)}$ , which is in fact a classical analytic norm, now keeping track of analyticity radius as an independent variable generated by the evolution of differential equations at hand.

Alternatively, for sake of convenience with algebra, we can introduce generator functions for analytic functions that are periodic in ${x\in \mathbb{T}^d}$ defined by

$\displaystyle Gen[u](z) = \sum_{\alpha \in \mathbb{Z}^d} |\widehat{u}_\alpha | e^{z |\alpha|} \ \ \ \ \ (1)$

for analyticity radius ${z\ge 0}$ , where ${\widehat{u}_\alpha}$ denote the Fourier coefficient of ${u(x)}$ with the corresponding Fourier variable ${\alpha}$ . Again, this is another classical analytic norm.

In case of the whole space ${x\in \mathbb{R}^d}$ or even domains with a boundary, similar generator functions can be introduced, replacing the above summation in ${\alpha}$ by integration 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, this paper of mine with Grenier for the case of boundary layers on the half space ${\mathbb{T} \times \mathbb{R}_+}$ .

Generator functions are non negative and control all derivatives of the function. In addition, all their derivatives are also non negative and non decreasing in ${z}$ . More conveniently, they enjoy some nice algebras (I shall only focus on the formulation (1); the other is similar). For instance,

$\displaystyle Gen [uv] \le Gen[u] Gen[v]$

which is direct, using Young’s inequality for convolution: namely, for ${z\ge 0}$ , we compute

$\displaystyle \begin{aligned} Gen[uv] &= \sum_{\alpha \in \mathbb{Z}^d} e^{z |\alpha|} |\widehat u_\alpha \star_\alpha \widehat v_\alpha | \le \sum_{\alpha \in \mathbb{Z}^d} \sum_{\beta \in \mathbb{Z}^d} e^{z |\beta|} e^{z|\alpha-\beta|} |\widehat u_{\beta}| |\widehat v_{\alpha-\beta} | \\&\le Gen[u] Gen[v] . \end{aligned}$

Importantly, they commute nicely with derivatives:

$\displaystyle Gen[\partial_xu] = \partial_zGen[u]$

which is again direct from the definition. These simple properties make generator functions very versatile to applications, which I shall now review a few.

2. An abstract framework

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

$\displaystyle \partial_t u = A(u, \partial_xu) \ \ \ \ \ (2)$

for unknowns ${u = u(x,t)}$ , where ${A(\cdot,\cdot)}$ satisfies

$\displaystyle Gen[A(u,\partial_x u)] \le C_0( 1 + \partial_z Gen[u]) F( Gen[u] ) \ \ \ \ \ (3)$

for some analytic function ${F(\cdot)}$ . For instance, ${A}$ may be of the form ${F(u)\partial_x u}$ , which covers many classical quasilinear PDEs in applications, including the classical Euler equations in fluid dynamics, hyperbolic equations in gas dynamics, Vlasov models in plasma physics, to name a few. Note however that no assumption on the hyperbolicity of the system is made (i.e. the system may be illposed in Sobolev spaces). The condition (3) is easily verified in these applications, using the mentioned properties of generator functions (i.e. algebra for product and commutation with derivatives).

Formally, applying generators to the evolution (2) and using the condition (3), one immediately arrives at a transport differential inequality for generators

$\displaystyle \partial_t Gen[u] \le C_0 ( 1 + \partial_z Gen[u]) F( Gen[u] ).\ \ \ \ \ (4)$

This differential inequality describes in an acute way how the radius of analyticity shrinks as time goes on, and allows one to get analytic bounds on ${u}$ , and in particular to bound all its derivatives at the same time. Indeed, introducing ${z = z(t)}$ so that

$\displaystyle z'(t) + C_0 F(Gen[u] (t,z(t))) \le 0\ \ \ \ \ (5)$

with ${z(0) = z_0 >0}$ , we get directly from the differential inequality (4) that along ${z=z(t)}$ dictated by (5),

$\displaystyle \frac{d}{dt}Gen[u](t,z(t)) \le C_0 F( Gen[u] (t,z(t)))$

which gives a priori estimates on ${Gen[u](t,z(t))}$ , and thus, on ${Gen[u](t,z)}$ for all ${z\in [0,z(t)]}$ as generator functions are non-increasing in ${z}$ . The existence of solutions ${u}$ follows. In practice, all these steps can be made rigorously via the standard approximation procedure: namely, first projecting the evolution onto finite Fourier modes so that generators are well defined through finite summation and deriving the above a priori estimates, which are uniform in the number of modes. The standard compactness argument gives a local-in-time analytic solution:

Theorem 1 Let ${\rho_0>0}$ and ${u_0}$ be in analytic function space ${X_{\rho_0}}$ defined by

$\displaystyle X_\rho : = \Big \{ u~:~ Gen[u](\rho) <\infty \Big\}.$

Then the Cauchy problem (2) 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 ${u_0}$ .

3. Euler equations

The previous framework straightforwardly applies to the classical incompressible Euler equations, which can be written as

$\displaystyle \partial_t u = - \mathbb{P} (u\cdot \nabla_x u),$

where ${ \mathbb{P} }$ is the classical Leray projector ${\mathbb{P} }$ , that is the projection onto the divergence-free ${L^2}$ vector space. In Fourier coefficients, ${\mathbb{P}_\alpha}$ is an ${d\times d}$ matrix with entries ${ (\mathbb{P}_\alpha)_{jk} = \delta_{jk} - \frac{\alpha_j\alpha_k}{|\alpha|^2}. }$ The local-in-time analytic solutions follow. Of course, this existence result is classical; see, for instance, C. Bardos and S. Benachour. The abstract framework can also be applied to a variety of other physical relevant models that arise in a singular limit of Euler equations, Navier-Stokes equations, and Vlasov-Poisson systems, where the limiting system often experiences loss of derivatives and may thus be illposed in Sobolev spaces; see our arxiv preprint and the references therein.

4. Landau damping

The abstract framework can also be used to provide analytic solutions at the large time as long as the analyticity radius ${z=z(t)}$ doesn’t shrink to zero at a finite time. For instance, we can obtain the celebrated nonlinear Landau damping for analytic and Gevrey data in this elementary way, since the electric field, which controls the shrinking of analyticity radius through (5), decays exponentially fast for these high regularity data. I have already blogged it here on this proof.

5. Boundary layers

Analyticity is a convenient way to obtain solutions, but often misses the rich underlying physics (e.g., singularity, instability, oscillation,…) which arises at high frequencies. In this section, I briefly mention another use of generator functions to capture some physics that is otherwise missed for analytic data.

Namely, consider the classical incompressible Navier-Stokes equations on the half-space in the small viscosity limit:

$\displaystyle \begin{aligned} \partial_t u + u\cdot \nabla u + \nabla p &= \nu \Delta u \\ \nabla \cdot u &=0 \end{aligned}$

with the classical no-slip boundary condition ${u=0}$ . In the inviscid limit ${\nu \rightarrow 0}$ , solutions formally converge to solutions of Euler equations with zero viscosity. However, due to the discrepancy of boundary conditions (for Euler, only normal component of velocity is imposed), boundary layers arise, which is known as classical Prandtl’s boundary layers:

$\displaystyle u_{NS}(t,x) = u_{Euler}(t,x) + u_{Prandtl}(t,x,\frac{x_n}{\sqrt \nu})+ O(\sqrt \nu)_{L^\infty}$

in the small viscosity limit ${\nu \rightarrow 0}$ . This classical Ansatz is rigorously justified for analytic data in the celebrated work of Sammartino-Caflisch ’98. It can also be justified for Gevrey data in some prescribed setting (Gerard-Varet–Maekawa-Masmoudi ’20). However, the Ansatz is false for Sobolev data, precisely due to the rich underlying physics which are not seen for analytic or Gevrey data.

To see the physics, one needs to go back to classical works by prominent physicists in the early of the last century such as Heisenberg, Tollmien, C.C. Lin, and Schlichting on hydrodynamic stability of shear flows. The assertion is that all shear flows, but Couette, are spectrally unstable at the high Reynolds number or the small viscosity limit, and there is an emergence of so-called Tollmien-Schlichting waves, which is onset of the laminar to turbulent transition (for a complete mathematical proof of this viscous destabilization, see this joint work of mine with Y. Guo and E. Grenier). This instability gives rise to another viscous boundary sublayer which is of a smaller thickness of order ${\nu^{3/4}}$ , as compared to the main classical thickness of Prandtl of size ${\nu^{1/2}}$ . It is this very sublayer that prevents the above classical Ansatz from being valid: namely, one would expect to have

$\displaystyle u_{NS}(t,x) = u_{Euler}(t,x) + u_{Prandtl}(t,x,\frac{x_n}{\sqrt \nu}) + u_{sub}(t,x,\frac{x_n}{\nu^{3/4}})+ O(\sqrt \nu)_{L^\infty}$

in the inviscid limit ${\nu \rightarrow 0}$ , for Sobolev data (it will be clear below that even this is false!). Note that as proved in the classical mentioned work of Sammartino-Caflisch for analytic data, this sublayer is absent.

Grenier was the first to rigorously capture this sublayer which can grow up to an amplitude of order ${\nu^{1/4}}$ (see this blog post where I discuss his approach). Recently, together with E. Grenier in this paper, we were able to prove that such a sublayer could reach to order one in its amplitude (i.e. non-vanishing in the inviscid limit). The importance of an order-one sublayer is that not only it disproves the classical Prandtl’s Ansatz, but also reveals a boundary layer cascade of smaller and smaller scales developed near the boundary that is completely missed for analytic data. Indeed, this sublayer itself again experiences its own Heisenberg’s viscous destabilization, since the local Reynolds number within this sublayer is of order ${\frac{u_{sub}}{\nu^{1/4}} \rightarrow \infty}$ as soon as the amplitude of sublayers ${u_{sub}}$ goes beyond ${\nu^{1/4}}$ . The instability gives rise to yet smaller and smaller sublayers. As a consequence, there are main Prandtl’s layers and many smaller sublayers, all of which are spectrally unstable! It is unclear which sublayers are dominant in the large time (we know one of them must be unstable!). This is a fundamental limitation to the previous approach.

Making use of generator functions, we are able to capture the order-one sublayer and avoid all smaller sublayers that may otherwise destabilize the instability mechanism. The capture of order-one sublayers not only disproves the classical Prandtl’s boundary layer Ansatz, but reveals the rich of underlying physics. Alright, as this blog post is to serve an introduction to generator functions, I skip but refer to this joint paper of mine with E. Grenier for further details as well as mentioned results on boundary layers.

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Generator functions and their applications

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