Generator functions and their applications

In [Grenier-Nguyen], we introduced so called {\em generators} functions to precisely follow the regularity of analytic solutions of Navier Stokes equations. In this short note, we give a presentation of these generator functions and use them to give existence results of analytic solutions to some classical equations, namely to hyperbolic equations, to incompressible Euler equations, and to hydrostatic Euler and Vlasov models. The use of these generator functions appear to be an alternative way to the use of the classical abstract Cauchy-Kovalevskaya theorem [Asano,Caflisch,Nirenberg,Safonov].


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], Caflisch [10], Nirenberg [20], Safonov [21], among others.In this short note, we use generators functions as introduced in [13] to obtain existence results for these equations.The results are not new, but show the versatility and simplicity of use of these generator functions.We believe that such approaches could be used on many other equations and provide easy ways to get existence results or to investigate instabilities.
Let us first introduce generator functions in the particular case of a periodic function f (t, x) on t ≥ 0 and x ∈ T d , d ≥ 1.For z ∈ R, we introduce the generator function Gen[f ] defined by in which f α denotes the Fourier transform of f (t, x) with respect to x ∈ 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 ∈ R d , with which the above summation is replaced by the integral over R d .In applications, we may also introduce generator functions depending on multi-variables , respectively; see, for instance, [13] for the case of boundary layers on the half space T × 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 for all z ≥ 0.
Proof.Let f α and g α be the Fourier transform of f and g, respectively.It follows that which is the second inequality.The first and third identities follow directly from the definition.The last inequality is a direct consequence of The use of generator functions are not limited to polynomial operations.Namely, we have Lemma 1.2.Let F be an analytic function with some convergence radius ρ.Let us introduce Then, for any function f , provided |f | < ρ, there holds ( Proof.First, using (1.3), we have The lemma follows from multiplying the inequality by z n and summing over n.
Moreover generator functions easily combine with truncation.Let P N be the projection of the first N Fourier modes; namely Then it follows that The main point is that if f (t, x) satisfies a non linear partial differential equation then Gen[f ] satisfies a similar differential inequality.This inequality describes in an acute way how the radius of analyticity shrinks as time goes on, and allows to get analytic bounds on f , and in particular to bound all its derivatives at the same time.We now illustrate this remark in the particular case of hyperbolic equations.

Hyperbolic equations
We consider the following system of evolution equations for a vector valued function u = u(x, t), with x ∈ T d , d ≥ 1, and t ≥ 0. We assume that the function A(•, •) satisfies for some constant C 0 and some analytic functions F .For instance, A may be a quadratic polynomial in u and ∇u, in which case F = Id, or more generally, A may be of the form We shall construct solutions in the function space X ρ defined by Theorem 2.1.Let ρ > 0 and u 0 be in X ρ .Then the Cauchy problem (2.4) with initial data u 0 has a unique solution u(t) in X ρ(t) for positive times t as long as ρ(t) = ρ − 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, recall the projection P N defined as in (1.7).Let u N (t) be a solution to the following regularized equations 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 ρ(t) , as long as ρ(t) remains positive.The Fourier coefficients of the solution to (2.4) satisfy where P N,α f denotes the Fourier coefficient f α if |α| ≤ N , and zero, if otherwise.We therefore get Using (1.8) and the assumption (2.2), we obtain the following Hopf-type differential inequality in which C 0 is independent of N .For convenience, we set The previous inequality yields 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 +∞, G N (0, z) converges to Gen[u 0 ](z), which is defined for |z| ≤ ρ.
As usual with analytic solutions, the domain of analyticity shrinks with time, hence we introduce for t, z ≥ 0, where θ(•) will be determined later.It follows that F N satisfies (2.7) Note that F N is defined for any z.In the limit N → +∞ we focus on 0 ≤ z ≤ ρ.
We will choose θ(t) is such a way that the characteristics of (2.7) are outgoing on [0, ρ], namely such that at z = 0, F (F N ) > 0 (which is always satisfied) and such that at z = ρ, F (F N ) + θ ′ (t)ρ < 0.
At t = 0, we choose θ(0) = 1 and thus We will focus on times 0 ≤ t ≤ T N such that F N (t, z) ≤ 2M 0 for 0 ≤ t ≤ T N and 0 ≤ z ≤ ρ.We choose Observe that on 0 ≤ t ≤ T N , as long as θ(t) > 0, F (F N ) + θ ′ (t)ρ < 0. On such a time interval, (2.7) is a nonlinear transport equation with a source term and with outgoing characteristics at 0 and ρ.As a consequence, we have Classical arguments then leads to the fact that T N is bounded away from 0, namely there exists some T > 0 such that T N ≥ T for any N , and such that ρ(t) > 0 for any t ≤ T .This implies that for any 0 ≤ z ≤ ρ and any 0 ≤ t ≤ T .Thus u N (t) is uniformly bounded in X ρ(t) for all N .As a consequence, u N and all its derivatives of all orders are uniformly bounded in L ∞ .Up to the extraction of a subsequence, u N and all its derivatives converge uniformly, towards some function u, solution of (2.4).Moreover, classical arguments show that u(t) ∈ X ρ(t) for any 0 ≤ t ≤ T , which ends the proof of the theorem.

Euler equations
In this section, we apply the previous framework to construct analytic solutions to incompressible Euler equations on T d or R d , d ≥ 2. Namely, we consider on T d (the case R d is treated similarly).Again, the existence result is classical (see, for instance, [4,5]).The function space X ρ is defined in (2.3).We have Theorem 3.1.Let ρ > 0 and u 0 be a divergence-free vector field in X ρ .
Then the Cauchy problem (3.1) with initial data u 0 has a solution u(t) in X ρ(t) for positive times t as long as ρ(t) = ρ− C 1 t remains positive, C 1 being a constant depending on u 0 .
Proof.Introduce the Leray projector P, projection onto the divergence-free L 2 vector space.In Fourier coefficients, P α is an d × d matrix with entries In particular, |P α | is bounded, uniformly in α ∈ Z d .Taking the Leray projection of (3.1), we obtain which falls into the previous abstract framework.It remains to check the assumption (2.2).Indeed, using (1.2) and the uniform boundedness of |P α | in α, we compute Theorem 3.1 follows from Theorem 2.1, upon noting that the divergence-free condition is invariant under (3.2).

Hydrostatic Euler equation and Vlasov models
In this section, we apply the abstract framework to two more examples that arise as a singular limit of Euler equations and Vlasov-Poisson systems.The first is the hydrostatic Euler equations which read The model (4.1)-(4.3)can be derived from the two-dimensional incompressible Euler equations in a narrow channel [19,12,9].The Cauchy problem for data with Sobolev regularity is illposed [15], except for convex profiles [8], due to the loss of derivatives in x through However, the local solution can be constructed for data with analytic regularity [18].
We now show that the previous abstract framework also applies to the hydrostatic Euler model (4.1)-(4.3).For z ≥ 0, we introduce the notion of generator functions as follows: in which ω α (y) denotes the Fourier transform of ω(x, y) with respect to x.The analyticity radius for x and y needs not to be the same [13], however for simplicity, we have taken the generator of the form (4.4).We obtain the following.
Proof.Let us first check the properties of the new generator function.For any f, g, we compute for any α ∈ Z and β ≥ 0. Therefore, for z ≥ 0, we have which proves the second identity in Lemma 1.1.The other identities are straightforward.
Let us now verify assumption (2.2).In this case where u = ∇ ⊥ ϕ, with ϕ defined through (4.2)-(4.3).We first note that Next, ∂ 2 y ϕ = ω (with the Dirichlet boundary conditions on [−1, 1]) can be explicitly solved, leading to This yields to recalling the loss of derivatives in variable x through u 2 = −∂ x ϕ.As a result, we have obtained To fit into the previous framework, we introduce the vector functions It follows that ∂ t U = A(U, ∇U ).The theorem follows from the previous abstract framework.
We conclude our paper with an application to kinetic models, known as the kinetic incompressible Euler and Vlasov-Dirac-Benney systems.These models are derived from the quasineutral limit or the vanishing Debye length limit of Vlasov-Poisson systems [7,11,14,16,2,15].The Vlasov equation reads on T d × R d , d ≥ 1, where the potential function ϕ is defined by for the Vlasov-Dirac-Benney system [2,14], or alternatively, defined through the elliptic equation for the kinetic incompressible Euler model [7].Again, the system (4.6)-(4.8)experiences a loss of derivatives in x.As a consequence, the Cauchy problem for data with Sobolev regularity is illposed [15], except for stable data [3,16].The local analytic solutions were constructed in [14,17,6].We now show that the abstract framework also applies to the kinetic model (4.6)-(4.8).Indeed, we introduce the following modified notion of generator functions: for some m > d + 2, in which f α (v) denotes the Fourier transform of f (x, v) with respect to x, and v = (1 + v 2 ) 1/2 .With exception of the weight in v, the generator function is identical to (4.4).Thus, Lemma 1.1 follows similarly.As a result, we have the following Again, as in the previous case, to fit into the abstract framework, we introduce the vector functions which yields ∂ t U = A(U, ∇U ).The theorem follows from Theorem (2.1).