The Cauchy problem and integrability of a modified Euler-Poisson equation

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in $H^{s} (T^{m})$ when $s>m/2+2$ and we improve the Sobolev index to $s>3/2$ for $m=1$. We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space $ Diff \ltimes C^{\infty}(\tor)$ as a Hamiltonian equation, we concentrate to one space dimension ($m=1$) and show that the equation is bihamiltonian.

(mEP) as well as its hamiltonian structure and integrability. The equation (mEP) is related to the Euler-Poisson equation which describes the fluctuations in the ion density of a two-component plasma of positively charged ions and negatively charged electrons (therefore it is also called ion acoustic plasma equation [LiSat]).
Many different techniques have been developed based on Picard's contraction theorem on Banach spaces in the study of nonlinear partial differential equations. One approach originated in an observation of V. Arnold [Arn] that the initial value problem for the classical Euler equations of a perfect fluid can be stated as a problem of finding geodesics on the group of volume preserving diffeomorphisms. Subsequently, this observation was used by D.G. Ebin and J. Marsden in [EMa] who developed the necessary functional analytic tools and established sharp local well-posedness results for the Euler equations in a class of Sobolev spaces.
The first section of this work is devoted to develop an appropriate analytic framework for the modified Euler-Poisson equation (mEP) using a similar approach and prove the following theorem.
Theorem 1. For s > m/2+1, given any initial data (n 0 , v 0 ) ∈ H s−1 (T m , R)× H s (T m , R m ), there exists a T > 0 and a unique solution (n, v) to the Cauchy problem for the modified Euler-Poisson equation (mEP) and and the solution (n, v) depends continuously on the initial data (n 0 , v 0 ).
Another powerful tool in the study of partial differential equations is the Cauchy-Kowalevski theorem. An abstract version of this theorem was developed by L.V. Ovsjannikov [Ovs1,Ovs2], F. Treves [Tre], L. Nirenberg [Nir], T. Nishida [Nis] and M.S. Baouendi and C. Goulaouic [BG] among others and subsequently applied to the Euler and Navier-Stokes equations.
The study of analytic regularity of solutions to the Camassa-Holm equation by A. Himonas and G. Misio lek [HM1], [HM3] using this abstract theorem led us to investigate the analytic regularity for (mEP). In Section 2, we prove the existence and uniqueness of local analytic solutions to the Cauchy problem for the equation (mEP).
Theorem 2. If the initial data (n 0 , v 0 ) is analytic on T m × T m then there exists an ε > 0 and a unique solution (n, v) of the Cauchy problem for the equation (mEP) that is analytic both in x and t on T m × T m for all t in (−ε, ε).
This result can be viewed as a Cauchy-Kowalevski type result for the equation (mEP). Even though the equation (mEP) admits an approximation by the Korteweg-De Vries equation, the analytic regularity results for the two equations are quite different. In contrast with the Korteweg-De Vries equation whose solutions are analytic in the space variable for all time but not analytic in the time variable (see [Tru], [KaM]), the solutions to the modified Euler-Poisson equation are analytic in both space and time variables.
In the third section we derive the equation (mEP) as a Hamiltonian equation on the semidirect product space Diff(T m ) ⋉ C ∞ (T m ) following the treatment of V. Arnold and B. Khesin in [AK] and J. Marsden, T. Ratiu and A. Weinstein in [MRW] of the Hamiltonian formalism related to fluid and gas dynamics. Then we concentrate to the one space dimension m = 1 and prove the following theorem.
Theorem 3. For m = 1 the modified Euler-Poisson equation (mEP) is bihamiltonian with the pair of hamiltonian functionals In the proof we use prolongations to check the compatibility of the induced Poisson brackets by these hamiltonian structures. In particular the modified Euler-Poisson equation (mEP) can be derived as a Hamiltonian equation on the semidirect product space of the Virasoro algebra with the smooth functions on the torus vir ⋉ C ∞ (T) along with a nonlocal hierarchy of equations called Hunter-Zheng equations (see [BDP] for the bihamiltonian structure of the Hunter-Zheng equations).
Remark 1. The Korteweg-de Vries equation (KdV) can be derived as an approximation to the Euler-Poisson equation by a perturbation analysis (see [Sat]). Using this approach it is straightforward to obtain an approximation to the system of equations in (2) which preserves the dispersion and leads to KdV.

Local well-posedness in Sobolev spaces
In this section we study the Cauchy problem for the modified Euler-Poisson In order to prove Theorem 1 we use the method of first restating the problem as an initial value problem for an ordinary differential equation on the group of diffeomorphisms of Sobolev class H s and then applying the existence theorem for vector fields on Banach manifolds.
Therefore the Cauchy problem for (mEP) can be reformulated as an initial value problem for the ordinary differential equation where with initial data (γ 0 , η 0 , ζ 0 ) = (id x , v 0 (x), n 0 (x)).
In the proof of Theorem 1 we repeatedly use three standard results on Sobolev spaces: The Schauder ring property, Sobolev imbedding theorem (that we refer to as Sobolev lemma) and the composition lemma (see, for example, [S] and [Ad]).
Proof of Theorem 1. If the map is locally Lipschitz then by the fundamental theorem for ordinary differential equations on Banach spaces [Di] there is a unique solution to the problem (4) for s > m/2 + 1 with initial data Note that the dependence of the solution of problem (4) on initial data is smooth. However the map γ → γ −1 on D s is continuous but not C 1 . Therefore we only have continuous dependence on initial data of the solution to the Cauchy problem for (mEP).
By Proposition 1 the proof of Theorem 1 is reduced to showing that the maps are locally Lipschitz in γ, η and ζ (uniformly with respect to the remaining variables). In the following estimates, the subscripts γ and ζ of a constant indicate the dependence of the constant on γ H s and ζ H s−1 respectively.
Then it is enough to show that the following estimate holds: Next we show that (6) holds for s > m/2 + 1. The left side of the inequality (6) has the following form We first estimate the L 2 term in (7). For any r > m/2 + σ we have the Sobolev imbedding into a Hölder space for any x, y ∈ T. Therefore using the composition lemma and applying (9) with r = s − 1 and σ = (s − 1 − m/2)/2 we find Adding and subtracting the appropriate terms we estimate the H s−1 term in (8) by the sum Using Schauder ring property and composition lemma the first summand in (10) is bounded by In order to estimate the second summand (11) we add and subtract the terms Λ −2 (ζ • γ −1 ) • γ and Λ −2 ζ. After cancellations we obtain Let u be Λ −2 (ζ • γ −1 ). Then we have Using the Sobolev lemma with the composition lemma we obtain the estimate for (12). For s > m/2 + 2 the term (13) can easily be estimated like (12). For m/2 + 1 < s ≤ m/2 + 2 we first observe that the estimate holds for s − 3 ≤ 0 and then using (9) as before, we obtain the following estimate for (13) where σ is equal to (s − 1 − m/2)/2 > 0. However the assumption s − 3 ≤ 0 does not follow from s ≤ m/2 + 2 if m ≥ 3. Nevertheless one can use the following inductive argument until s − (2k + 1) ≤ 0 (it ends in finitely many steps since s ≤ m/2 + 2). If s − 3 > 0 we split (13) as in (7)-(8). The L 2 part can be estimated as is bounded by Here the first summand (18) is estimated using the Schauder ring property with the composition lemma For the second summand (19) we use the steps (6)-(14) to reduce it to estimating If s − 5 ≤ 0 we proceed as in (15)-(16). Otherwise we repeat the steps (17)-(20).
. Then by Schauder ring property we have Using the Schauder ring property one more time we bound this term by and therefore γ → F (γ, ζ, η) is locally Lipschitz. It is straightforward to show that the second, third and fifth maps in (5) are uniformly Lipschitz using properties of Sobolev spaces. This completes the proof of Theorem 1.
Next we observe that the Cauchy problems for the equation (mEP) and the Euler equations of an incompressible fluid are not only similar for low regularity (Sobolev class) data but also for high regularity (analytic) data.

Analytic regularity
In this section we give a proof of theorem 2 that states the analytic regularity (i.e., existence and uniqueness of analytic solutions for analytic initial data) of the Cauchy problem for (mEP).
Our approach is motivated by the work of M.S. Baouendi and C. Goulaouic [BG] who studied analytic regularity of the Cauchy problem for Euler equations of incompressible fluids.
The proof of theorem 2 relies on a contraction argument in a decreasing scale of Banach spaces X s (i.e. if s ′ < s implies X s ⊂ X s ′ and ||| · ||| s ′ ≤ ||| · ||| s ).
For s > 0, let the spaces E s be defined as where σ is any integer such that σ > 1 + m/2 and let X s be given by the Cartesian product E s × E s . The norm ||| · ||| Xs can be chosen to be any of the standard product norms on E s × E s . The following lemma states the ring property for the spaces E s . Lemma 1. Let 0 < s < 1. There is a constant c > 0 which is independent of s such that we have |||uv||| s ≤ c|||u||| s |||v||| s for any u, v ∈ E s .
Lemma 2. For 0 < s ′ < s < 1, we have Proof. By the definition of |||.||| s , we have The H σ norm on the right hand side can be written in the local coordinates up to a constant as Then we have the estimates Note also that sup |β|=1 (k + β)! k! = sup 1≤i≤m (k i + 1) ≤ |k| + 1.
Lemma 3. For any 0 < s < 1, the estimate Proof. We write P 4 (u)v in terms of the linear operator Du as P 4 (u)v = (Du)v. Then by Lemma 1 we have Now we reduce the proof to the case that we handled in the proof of lemma 2: Clearly, to finish the proof, it is enough to show that Let s ′ = λs, 0 < λ < 1 and f (λ) = (k + 1)(1 − λ)λ k . Then, For k = 0 it is clear that f (λ) ≤ 1. For k ≥ 1 the function f (λ) = (k + 1)(1 − λ)λ k is continuous in the interval 0 < λ < 1 and it has zeros at the endpoints of the interval [0, 1] and a maximum at λ = k k+1 such that f ( k k+1 ) = k k+1 k < 1. Then we have and the formula (24) holds. Using the formula (24) we obtain the desired estimate Now we are ready to prove theorem 2. proof of theorem 2. We refer to the version of the abstract Cauchy-Kowalevski theorem in [Nis]. We only need to verify the first two conditions of this theorem since the map F (u 1 , u 2 ) does not depend on t explicitly.
By Lemma 2 and Lemma 1, we have Similarly, for F 2 , using the lemmas 2, 3 and 4 we have We proceed to establish the second condition of the abstract Cauchy-Kowalevski theorem. We will show that for some c independent of t, To obtain the first estimate above, after applying Lemma 2, we add and subtract the term u 1 v 2 and use Lemma 1: Then, assuming that |||u||| s < R and |||v||| s < R, we have To estimate the F 2 component, we use lemmas 2 and 3: Note that Using Lemma 4 and the above identity, (25) implies Therefore the estimate holds. This completes the proof of theorem 2.

Bihamiltonian structure and integrability
A number of partial differential equations that describe fluid motion can be derived as equations for geodesics on various infinite dimensional Lie groups. For instance, the Euler equation for ideal incompressible fluid flow is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold M with a right invariant metric given by the L 2 inner product on the tangent space at the identity of the group [EMa]. Other examples are • Korteweg-de Vries equation and Camassa-Holm equation on the Bott-Virasoro group (see for example [OK] and [Mis3]), • ideal incompressible MHD (magnetohydrodynamics) on the semidirect product of volume preserving diffeomorphisms with the divergence free vector fields, • Hunter-Saxton equation on the homogeneous space of all diffeomorphisms of the unit circle modulo the rotations [KM], etc.
In contrast with all the examples we gave above, the energy of the modified Euler-Poisson equation (mEP) is not a quadratic form, therefore it can not be interpreted as a Riemannian metric. However, there still is a variational problem on the cotangent space of the configuration space of this equation. Here we derive the equation (mEP) from this variational problem. Note that all the computations that follow are formal.
Let g be a Lie algebra with the bracket operation [·, ·] and g * be its dual given by the pairing ·, · : g * × g → R.
Then g * with the Lie-Poisson bracket defined by is an equivalent formulation of the Hamilton's equation on a Lie-Poisson manifold.
Here we exploit the tools and techniques used to study the Hamiltonian formulation of the Euler equations for a compressible fluid [MRW] to show that the modified Euler-Poisson equation (mEP) can be derived as a Hamiltonian equation.
On the Cartesian product space Diff(T m ) × C ∞ (T m ) of the group of diffeomorphisms of T m and the vector space C ∞ (T m ) of all smooth functions on T m , the operation called the semidirect product induces a Lie group structure. We denote this group by G = Diff(T m ) ⋉ C ∞ (T m ) following the conventional notation for semidirect product spaces. The corresponding Lie algebra is the space g where v, w ∈ Vect(T m ) and a, b ∈ C ∞ (T m ). Here [v, w] is the usual commutator of vector fields on T m and L w a is the Lie derivative of a in the direction of w and is given by L w a = d ds s=0 (a • ζ s ) where ζ s is any curve on Diff(T m ) such that ζ s | s=0 = id and d ds s=0 ζ s = w. Note that in this setting the composition of the diffeomorphisms is the group operation on Diff(T m ), the composition of a smooth function with a diffeomorphism a • γ −1 is the natural action of the diffeomorphism γ on the function a. In general, the semidirect product structure on the Cartesian product of a Lie group and a vector space on which the group acts is defined using the group operation and the action of the group on the vector space (see [AK], [MRW]).
In this context, a Hamiltonian formulation of the modified Euler-Poisson equation (mEP) can be stated as follows: Theorem 4. The modified Euler-Poisson equation (mEP) is a Hamiltonian equation on g * with respect to the linear Lie-Poisson structure and the energy function where M = nv ∈ Vect(T m ) and Φ ′ (n) = Λ −2 (n).
Proof. We want to derive the equations for v and n from where m = (M, n) = (nv, n) and (v, n) ∈ g * . The variational derivative δH δm is given by δH δM , δH 1 δn with Evaluating equation (29) on an arbitrary pair (w, b) ∈ g, we obtain Then by the definition of the coadjoint operator ad * and the bracket on g, we have In what follows we identify the dual space g * with the algebra g using the pairing ·, · on g * × g given by Then substituting nv for M and using (30) and (31), we obtain By the definition of the bracket [·, ·] on Vect(T m ), we have Furthermore, we can compute the Lie derivatives on the right hand side and write the above equality as follows Integrating by parts the first and the last summands on the right hand side, we obtain Then by (32), we have ∂ t n = −div(nv), which is equivalent to Note that for one space dimension (m = 1) the hamiltonian H 1 in (28) is given by in terms of v and n. Then using the differential operator D 1 that is defined as one can rewrite the equation (mEP) in the hamiltonian form The Poisson bracket induced by the matrix differential operator D 1 is given by Another conserved quantity for (mEP) is For one space dimension (m = 1) we can use H 2 to write (mEP) in yet another form as ∂ t v n = D 2 δH 2 /δv δH 2 /δn where D 2 is defined as We prove theorem 3 by showing that (34) and (36) are Hamiltonian forms of the modified Euler-Poisson equation (mEP) and that the induced Poisson structures are compatible. proof of theorem 3. The matrix differential operator D 1 is skewadjoint and does not depend on v nor n nor any of their derivatives, therefore the bracket given by D 1 satisfies the Jacobi identity hence is indeed a Poisson bracket.
We can easily check that D 2 is skew-adjoint as well: (φ 1 , φ 2 )D 2 (θ 1 , θ 2 )dx To verify the Jacobi identity for the bracket induced by D 2 we adapt the notation of prolongations (see [Ol] for details). Let Θ 2 be the functional bivector associated to D 2 : Then D 2 is Hamiltonian since These two Hamiltonian structures, (34) and (36), are compatible, i.e. the equation (mEP) is bihamiltonian in one space dimension. To prove the compatibility it is enough to check that pr v D 1 θ (Θ D 2 ) + pr v D 2 θ (Θ D 1 ) = 0 (37) holds where Θ D i denotes the corresponding bivector for D i . Both summands in (37) vanish: and similarly we have Therefore the modified Euler-Poisson equation (mEP) is bihamiltonian for m = 1 and this completes the proof of theorem 3.