On the existence of chaos for the viscous van Wijngaarden–Eringen equation

. We study the viscous van WijngaardenEringen equation:


Introduction
In the 1940's and 50's, the interest in studying the propagation of pressure waves of small amplitude in bubbly liquids appeared.The reason was to determine whether it was possible to take advantage of these acoustical properties to control the sound produced by propellers, both of surface ships and submerged ship.A vast literature on the subject deals with theoretical and experimental studies of the various aspects of propagation of pressure waves of small amplitude in bubbly liquids, see for instance [?].
The acoustic planar propagation perpendicular to and along the x-axis (i.e., 1D flow) in bubbly liquids is given by the following equation [?, Eq. 4.14] (2) ∂ 2 u ∂t 2 − (1 − 2ε(β − 1)) being Re d = c e L/δ a Reynolds number, where c e (> 0) is the adiabatic sound speed, δ is the diffusivity of sound [?], and L is a characteristic (macroscopic length).The constant a 0 is a Knudsen number that corresponds to the dimensionless bubble radius.In addition, γ stands for the adiabatic index of the liquid and β(> 1) is known as the coefficient of nonlinearity [?].This coefficient is given by β = (γ + 1)/2 in the case of a perfect gas.
More details on the formulation of equation (2) can be found in [?] and [?].The linearized version, ε = 0, of equation ( 2) is known as the viscous (or dissipative) van  (3) Many problems in acoustics for the sound propagation are described in terms of the linear wave equation.The transport equation governing the propagation of second-sound in both metals and dielectric solids is the well known damped wave equation [?, ?] (see also [?] for recent advances in the area).For these two cases, the 1D version of this PDE coincides with the hyperbolic heat transfer equation.The chaotic behaviour of the solutions of the abstract Cauchy problem stated with this equation was investigated in [CPT10], see also [GEPM11].For high wave amplitudes and intensities, phenomena such as wave distortion and formation of shocks appear, and then it is natural to study chaos for this type of equations.In [CLR15] the chaotic behaviour of the one-dimensional version of the Moore-Gibson-Thompson equation is studied.See also the seminal formulation of this equation in [?, Eq. 7].Moreover, the strong connection between acoustics and bioengineering and industry of high intensity sound waves has contributed to the improvement of the research in this area (see [?], [?], [?]).
To the best of our knowledge, no study on chaotic behavior for equation (1) has been carried on.Hence, our aim here is to examine the van Wijngaarden-Eringen equation in the context of a dynamic, yet still analytically tractable, setting.
In this paper, we succeed in proving the existence of a chaotic dynamics for the van Wijngaarden-Eringen equation.More precisely, we are able to show that whenever a 0 < 1 and (4) then equation (1) admits a uniformly continuous semigroup which is Devaney chaotic on an isomorphic copy of the sequence space c 2 0 (N 0 ).The paper is organized as follows.In section 2 we introduce some basic concepts related to the study of C 0 -semigroups and chaos.In Section 3, chaos is also studied for the viscous van Wijngaarden-Eringen equation.Moreover, we provide a range for the bubble radius that ensures the existence of chaotic behaviour for the solutions of this equation.Finally, in Section 4, we give some physical interpretation of the results proved previously.

Preliminaries
Let X be a separable infinite-dimensional Banach space.We recall that {T t } t≥0 , with T t : X → X a continuous and linear map on X for each t ≥ 0, is a C 0 -semigroup if T 0 = I, T t+s = T t • T s and lim s→t T s x = T t x for all x ∈ X and t ≥ 0.
Let {T t } t≥0 an arbitrary C 0 -semigroup on X.It can be shown that ( 5) exists on a dense subspace of X; the set of these x, the domain of A, is denoted by D(A).
Then A, or rather (A, D(A)), is called the infinitesimal generator of the semigroup.
Conversely, the Hille-Yosida theorem states that, under suitable assumption on a densely defined operator (A, D(A)), the solutions of the following abstract Cauchy problem on X: , then the operators in the C 0 -semigroup can be represented as hypercyclic and the set of periodic points is dense in X.We point out that these two requirements also yield the sensitive dependence on the initial conditions, as it was seen by Banks et al [BBC + 92, GEPM11].Further information on the dynamics of C 0 -semigroups can be found in [GEPM11,Ch. 7].See also [Eis10] for information regarding the stability properties of C 0 -semigroups.
Another variation of the definition of chaos is the notion of distributional chaos introduced by Schweizer and Smítal [SS94], see also [MGOP09,Opr06] for its presentation in the infinitedimensional linear setting.A C 0 -semigroup {T t } t≥0 on X is said to be distributionally chaotic if there exists an uncountable subset S ⊂ X and δ > 0 such that, for each pair of distinct points x, y ∈ S and for every ε > 0, we have Dens({s ≥ 0; ||T s x − T s y|| > δ}) = 1 and Dens({s ≥ 0; ||T s x − T s y|| < ε}) = 1, where Dens stands for the upper density of a set of real positive numbers.The set S is called the scrambled set and the C 0 -semigroup is said to be densely distributionally chaotic if S is dense on X.Now, we present a criterion that ensures Devaney chaos for C 0 -semigroups.It is a variation of the (DSW) criterion [DSW97] which depends on verifying that the point spectrum of the infinitesimal generator of the C 0 -semigroup contains "enough" eigenvalues.A first criterion stated in these terms was given for operators by Godefroy and Shapiro in [GS91].We will use the following version of the (DSW) Criterion, see [GEPM11,Th. 7.30].It is also well known that distributional chaos holds whenever the DSW criterion can be applied [BC12,BBMGP11].
A Borel probability measure (µ, B) is said to have full support if for all non-empty open set U ⊂ X we have µ(U ) > 0, and µ is said to be T t -invariant if for all A ∈ B we have that µ , for any A, B borelian sets of X.The existence of these measures are ensured if hypothesis of the DSW criterion are satisfied [MAP15].
Finally, we recall the definition of the space of analytic functions of Herzog type.Given ρ > 0, let: endowed with the norm f ρ = sup n≥0 |a n |.This space is isometrically isomorphic to c 0 (N 0 ).For examples and references on Herzog spaces, we refer the reader to [CLR15].

Existence of chaotic behavior
In this section, we will study the chaotic behavior of the viscous van Wijngaarden-Eringen equation in the space of analytic functions of Herzog type.We recall that a 0 > 0 denotes the dimensionless bubble radius and Re b is a Reynolds number.Since the second order differential operator ∂ xx turns out to be a bounded operator on X ρ , assuming the condition (8) a 0 < 1, we obtain a 2 0 ∂ xx ρ < 1 and, consequently, the inverse operator (1 − a 2 0 ∂ xx ) −1 exists on X ρ .Then we can express (7) as a first order equation on the product space X := X ρ ⊕ X ρ .Setting u 1 = u and u 2 = ∂u ∂t we can pose the following abstract Cauchy problem: Then, the operator-valued matrix (10) bounded operator on X and, consequently, we have that {e tA } t≥0 is the solution C 0semigroup of (9).See also [CPT10, Sec.2.2] for a similar representation of the solution in the case of the wave equation, and [CLR15, Sec.3] for the case of the Moore-Gibson-Thompson equation.
The following theorem is the main result in this paper.
Theorem 3.1.Suppose that a 0 < 1 and then for each ρ satisfying , the operator A generates a uniformly continuous semigroup which is Devaney and distributionally chaotic, topologically mixing and admits a strongly mixing measure with full support on X ρ ⊕ X ρ .
It only remains to show that for any x * ∈ X * ρ ⊕ X * ρ the functions λ → f z 0 ,z 1 (λ), x * , z 0 , z 1 ∈ C, are holomorphic on U , and if they all vanish on U , then x * = 0. Since X ρ is isometrically isomorphic to c 0 , in what follows, we identify the dual space X * ρ with 1 .Let x * ∈ X * ρ ⊕ X * ρ .It can be represented in a canonical way by for all λ ∈ U, z 0 , z 1 ∈ C.This last equation can be reformulated in the following way: Let λ 0 = 0.It is clear that λ 0 ∈ U and R λ 0 = 0. Evaluating (22) in λ 0 , we have the following equation: for all z 0 ∈ C. Therefore, x * 1,0 = 0. Now, we divide (22) by λ and we get: As R λ 0 = 0, evaluating (24) in λ 0 we obtain: for all z 0 , z 1 ∈ C.Then, x * 2,0 = 0 and x * 1,1 = 0. Now, we divide (22) by λ √ R λ and we get: So that equation (26) can be reduced to: Evaluating (27) in λ 0 , we have: 2,1 = 0 and x * 1,2 = 0. Proceeding inductively, we will get that x * i,n = 0 for i = 1, 2 and n ∈ N. We finally have x * = 0 and we conclude the result by applying Theorem 2.1.Remark 3.2.Recalling that a 0 > 0 denotes the bubble radius and Re b = c e L/δ is a Reynolds number, where L denote a characteristic length, c e the adiabatic sound speed and the positive constant δ > 0 is known as the diffusivity of sound, we observe that the condition (11), namely together with the general condition a 0 < 1, give an explicit range for the bubble radius in order to obtain chaotic behavior in the given model.This estimation can be compared with the physical one of (30) 0.4899 given in [?, Eq. 4.20] that has to be satisfied by the Reynolds number.

Conclusions
These results can be understood in the following sense.For the hypercyclicity, given an arbitrary acoustic planar wave close to the origin one can determine a wave far away enough in order that, as time goes by, we can have that its propagation through a bubbly liquid gives us a wave at the origin as close as we want to the initial one.The Devaney chaos also yields the existence of periodic waves similar as much as we want to a prescribed one close to the origin.
These theoretical results present some limitations: The amplitudes required to resemble a prescribed wave at the origin can be so wide that cannot even be easily generated.
Furthermore, we recall that the existence of Devaney chaos yields the sensitive dependence of the solutions to the problem respect to the initial conditions.This affirms that given some planar wave, one can find a small perturbation, such that after a long enough period of time the behavior of both waves is completely different.
One last comment, the existence of distributional chaos asserts, roughly speaking, that there is an uncountable set of initial conditions such that we can pick a pair of initial conditions from this set and, as time goes by, there will be long time intervals in which the behaviour of the waves close to the origin are very similar for both initial waves.On the other hand, there will be also intervals as long as the previous ones in which the waves are quite different depending on which one of these two initial conditions we have chosen, see [BCMASS15].
Some other classical models in nonlinear acoustics are the Kuznetsov equation, the Westervelt equation, and the Kokhlov-Zabolotskaya-Kuznetsov equation.Several initial boundary problems for these second order in time partial differential equations have been already solved (see for instance [?], [?], [?] and [?]).