On Stationary Nonequilibrium Measures for Wave Equations
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EMATICS
On Stationary Nonequilibrium Measures for Wave Equations T. V. Dudnikova Presented by Academician of the RAS B.N. Chetverushkin March 5, 2020 Received March 5, 2020; revised March 5, 2020; accepted March 23, 2020
Abstract—In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite mean energy density and study the convergence of distributions of the solutions to a limiting Gaussian measure for large times. We derive the formulas for the limiting energy current density (in mean) and find a new class of stationary nonequilibrium states for the studied model. Keywords: wave equations, random initial data, mixing condition, weak convergence of measures, Gaussian and Gibbs measures, energy current density, nonequilibrium states DOI: 10.1134/S1064562420030072
1. INTRODUCTION In the paper, we consider the wave equations in Rd (d ≥ 3 and odd) with constant or variable coefficients of the form d
u(x, t ) =
∂i (aij (x)∂j u(x, t )) − a0(x)u(x, t ),
i, j =1
x ∈R , d
(1)
i, j = 0,1,
u( x,0) = v0( x),
x, y ∈ Rd ,
have the form Q0ij ( x, y ) = q0ij ( x , y , x − y ), where x =
(x1, ..., xk), x = ( xk +1,…, xd ) , x = ( x , x ), y = ( y , y) ∈ Rd with some k ∈ {1,…, d} . Moreover,
t ∈ R,
Q0ij ( x, y ) = qnij ( x − y )
and with the initial data (as t = 0)
u( x,0) = u0( x),
Q0ij (x, y) = Y0i (x)Y0 j ( y)μ0(dY0),
for
x, y ∈ Dn ,
(3)
where the regions Dn are defined as follows: x ∈R . d
Dn = {x ∈ Rd : ( −1) 1 x1 > a, …, ( −1) k xk > a}, n
(2)
Here ∂ j ≡ ∂ , u( x, t ) ∈ R. We assume that the coeffi∂x j cients of the equation are sufficiently smooth, for x > R0 Eq. (1) has the form u( x, t ) = Δu( x, t ) ; a0( x) ≥ 0, and the matrix (aij ( x)) is positive definite for all
x ∈ Rd . In addition, we impose the so-called nontrapping condition (see condition D in [1]) which says that all rays of Eq. (1) go to infinity as t → ∞ . The initial data Y0( x) = (u0( x), v0( x)) are assumed to be a measurable random function with the distribution μ0. We assume that the correlation functions of the initial measure μ0, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia e-mail: [email protected]
n
n = ( n1, …, nk ) ∈ 1 k .
(4)
Here 1 k = {n = (n1,…, nk ): n j ∈ {1;2} , ∀j}, a is a fixed number, a > 0. Another words, condition (3) means n that in the case when ( −1) j x j > a for all j = 1,…, k , the random function Y0(x) is equal to different, general speaking, translation invariant random processes Yn(x) with distributions μn. Finally, we assume that the measure μ0 has a finite mean energy density,
Y
0
( x ) 2μ0 (dY ) = Q000 ( x, x ) + Q011 ( x, x ) ≤ C < ∞. (5)
Denote by μt, t ∈ R , the distribution of the solutions Y (t ) ≡ Y (,⋅ t ) = (u(,⋅ t ), u(,⋅ t )).
The main goal of the paper is to prove the weak convergence of the measures μt : 195
196
DUDNIKOVA
μt → μ ∞
as
t → ∞.
(6)
The similar convergence holds for t → −∞ since our
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