Invariance of the white noise for the Ostrovsky equation
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variance of the white noise for the Ostrovsky equation Mohamad Darwich1 Received: 24 February 2020 / Accepted: 27 August 2020 © Università degli Studi di Ferrara 2020
Abstract In this paper, we construct invariant measure for the Ostrovsky equation associated with the norm L 2 . On the other hand, we prove the local well-posedness in the besov space bˆ sp,∞ for sp > −1. Keywords Nonlinear wave equation · dispersive equations · Invariant measures Mathematics Subject Classification 35Axx · 35A01 · 37Kxx
1 Introduction In this paper, we construct an invariant measure for a dynamical system defined by the Ostrovsky equation (Ost) ∂t u − u x x x + ∂x−1 u + uu x = 0, (1.1) u(0, x) = u 0 (x) ∈ H s (T) where s ≥ − 21 and T = R/2π , for the quantities conserved by this equation. The operator ∂x−1 in the equation denotes a certain antiderivative with respect to the variable ˆ −1 f ) = f (ξ ) . x defined for 0-mean value periodic function the Fourier transform by (∂ x
iξ
Invariant measure play an important role in the theory of dynamical systems (DS). It is well known that the whole ergodic theory is based on this concept. On the other hand, they are necessary in various physical considerations. Solutions for the Ostrovsky equation was studied in [2]. Moreover, in [13] an infinite series of invariant measure associated with a higher conservation laws are constructed for the one-dimensional Korteweg de Vries (KdV) equation: u t + uu x + u x x x = 0,
B 1
Mohamad Darwich [email protected] Lebanese university, Hadath, Beirut, Lebanon
123
ANNALI DELL’UNIVERSITA’ DI FERRARA
by Zhidkov. In particular, invariant measure associated to the conservation of the energie are constructed for this equation. In the other hand, Tadahiro in [10] was construct an invariant measure for (KdV) equation associated to the conservation of norm L 2 using the notion of the Wiener space. Equation 1.1 is a perturbation of the Korteweg de Vries (KdV) equation with a nonlocal term and was deducted by Ostrovskii [11] as a model for weakly nonlinear long waves, in a rotating frame of reference, to describe the propagation of surface waves in the ocean. The natural conserved quantities for (1.1) are the : L 2 -norm: u(t) L 2 =
1 2
u (t, x)d x 2
and the Hamiltonian: 1 H (u(t)) = 2
1 (u x ) + 2 2
(∂x−1 u)2
1 − 6
u3.
We will construct invariant measures associated with the L 2 -norm using the notion of Wiener Spaces. The paper is organized as follows. In Sect. 2 the basic notation is introduced and the basic results are formulated. In Sect. 3 we give the precise mathematical meaning of the white noise Q defined an einx with mean 0 by for u = n
1 2 exp − |an | n≥1 dan n≥1 2 dQ = 1 2 |an | n≥1 dan exp − n≥1 2
(1.2)
and show that it is a (countably additive) probability measure on bˆ sp,∞ , p = 2+ , s = − 21 + such that sp < −1, defined via the norm ⎛ f bˆ s
p,∞
:= fˆbsp,∞ = supn fˆ(n) L p s
j
|n|∼2 j
= sup ⎝ j
⎞1
p
n | fˆ(n)| sp
p⎠
,
|n|∼2 j
and we go over the basic theor
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