Abstract
The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let \(m_0\) be the thermal conduction rest state and let \(P_r, P_m, R, Q\), be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of \(m_0\) via Hopf bifurcation can occur only in a plasma with \(P_m>P_r\), in this paper it is shown that the Hopf bifurcations occur if and only if
with \( \mu =7.8532\). Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be \(\displaystyle \frac{1}{4}\) of the previous value.
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1 Introduction
Let L be a horizontal layer heated from below, filled by a plasma and embedded in a transverse uniform magnetic field and let \(m_0\) denote the thermal conduction in L: the temperature field arising when the fluid is in rest and L is heated from below by a constant transverse gradient of temperature. The linear stability of \(m_0\)—in the scheme of the nonrelativistic thermal MHD—has been analyzed by Chandrasekhar in the early of 1950 (Chandrasekhar 1952, 1954) and it appeared in 1961 in the celebrated monograph (Chandrasekhar 1981). The Chandrasekhar results—for any type of boundaries (rigid–rigid, rigid–free, free–rigid, free–free)—have been since then a basic paradigm for all the subsequent researchers. In particular, he obtained a relevant inhibition of convection (stabilizing effect) by a magnetic field, verified successively experimentally (Nakagawa 1955). As concerns the occurring of the Hopf bifurcations at the onset of \(m_0\) instability, he restricted himself mostly to the case when the fluid is confined between two free planes, showing that bifurcations can occur only if \(P_m>P_r\) and envisaging the existence of a bound \(Q_c(P_r,P_m)>0\) such that if and only if
Hopf bifurcations occur. He wrote: “there is no simple formula which gives \(Q_c\) as function of \(P_r\) and \(P_m\)” (see Chandrasekhar 1981, p. 184). Recently, Rionero (2019a) has shown that in the free–free case, this difficulty can be removed and that the Hopf bifurcations occur if and only if
Let
be the spectrum equation governing the linear stability of a steady state of a dynamical system and let the coefficients depend on a positive bifurcation parameter R and on the wave number \(a^2\) of the perturbations in such a way that
with \({{\mathcal {F}}}_k(a^2)\) differentiable function of \(a^2\in {{\mathbb {R}}}^+\). Taking into account that the instability can occur only via a zero eigenvalue (\(\lambda =0\Leftrightarrow A_n=0\)), or via a couple of pure imaginary eigenvalues (\(\lambda _{1,2}=\pm i\omega \))—with i imaginary unit and \(\omega =\)const.\(\in {{\mathbb {R}}}^+\) such that \({{\mathcal {P}}}(i\omega )=0\)—the ideas that lead Rionero to the solution of the problem can be summarized as follows:
-
1.
the application of the critical numbers
$$\begin{aligned} R_{c_k}={{\mathcal {F}}}_k(a^2_{c_k})=\displaystyle \min _{a^2\in {{\mathbb {R}}}^+}{{\mathcal {F}}}_k(a^2) \end{aligned}$$(3)introduced by him, for \(k<n\), about 10 years ago (Rionero 2012, 2013, 2019a, b, 2020; Flavin and Rionero 1996);
-
2.
at the growing of R, from a stable state, the Hopf bifurcations occur if and only if exists at least a \(k<n\) such that \(R_{c_k}<R_{c_n}\), i.e. \(A_k\) becomes zero before \(A_n\): this is because—as it is well known—\(A_k>0,\,\forall k\in \{1,2,\ldots , n\}\), is necessary for the linear stability, i.e. for letting all the eigenvalues have negative real part;
-
3.
if the coefficients depend also on other parameters \(Q, P_m,P_r\), then one has \(R_{c_k}=R_{c_k}(Q,P_m,P_r)\) and the Hopf bifurcations can occur if and only if the inequality
$$\begin{aligned} R_{c_k}(Q,P_m,P_r)<R_{c_n}(Q,P_m,P_r), \end{aligned}$$for at least a \(k<n\), holds.
The free–free layers—of relevant interest in astrophysics applications—do not have the same interest in terrestrial—geophysics and industrial—applications when rigid boundary planes occur. In the present paper, aimed to remove the difficulty remarked by Chandrasekhar, in the presence of rigid boundary, we return to look for the Hopf bifurcations threshold in plasma layers between rigid planes, electricity perfectly conducting, via the guidelines (1)–(3).
In the case, at stake, the spectrum equation is
with the coefficients such that
with \({{\mathcal {F}}}_k\) differentiable function of \(a^2,P_m,P_r,Q\). Choosing—as it is standard in the convection problems—R as bifurcation parameter, via (1)–(3) the properties (1)–(3) will be proven in the in the rest of the paper, with \(A_2\) and \(A_3\) given in (51) and (52).
Property 1
Let \(m_0\) be linearly stable, for any wave number, at \(R=0\). Then, if and only if
an Hopf bifurcation occurs and occurs at \(R_*\in ]0,R_{c_2}[\) lowest root of
with \(a^2=a^2_{c_2}\) and has the frequency
It remains to satisfy (6). In other words, since \(P_m\) and \(P_r\) are structural parameters characterizing the fluid, the question is: in which fluids the Hopf bifurcations occur and for which values of Q? The answer is given by the following property.
Property 2
(Main result) If and only if
\(R_{c_2}<R_{c_3}\) holds.
Property 2 reduces the linear stability of \(m_0\) to a standard type of stability in convection problem. In fact, setting
one has
Property 3
The steady state \(m_0\) is linearly stable if and only if \(P_m<P_r\) and
or
or
The plan of the paper is the following. Section 2 is devoted to preliminaries. After having recalled the \(m_0\) thermal conduction rest state, the nonlinear MHD equations governing the perturbations to \(m_0\) are introduced. In Sect. 3, the linear stability of \(m_0\) is introduced and it is shown that it depends on the behaviour of the essential variables \(w, h, \theta \) with w, h vertical perturbations to velocity and magnetic fields, respectively, and \(\theta \) perturbation to the thermal field. The boundary conditions for \(w,h,\theta \), in the presence of rigid perfect conductor boundary planes, are considered in Sect. 4. In Sect. 5, the functional spaces of \(w,h,\theta \)—admissible by the boundary conditions—are introduced and the Sturm–Liouville type problem associated with w is taken into account: w can be expanded either in a Fourier series of even or odd functions. In Sect. 6, it is shown that \(m_0\) is linearly stable with respect to the perturbations \((w,h,\theta )\) with w even function. Successively, in Sect. 7, w odd is taken into account and the spectrum equation for the nth (\(n\in {{\mathbb {N}}}\)), Fourier components of the perturbations, is obtained. Property 1 is obtained in Sect. 8, while the other two are obtained in the subsequent section. The onset of Hopf bifurcations in the free-rigid and rigid-free cases is studied in Sect. 10. Final remarks are given in Sect. 11. The paper ends with an Appendix (Sect. 12) with three subsections concerning the invariants of \(3\times 3\) matrices, the Hurwitz criterion and the linear and nonlinear energy decay.
2 Preliminaries
Misregarding the displacement currents, the equations governing the thermal MHD under the validity of the state equation for the density \(\rho \),
are (see Chandrasekhar 1981, chapter 5)
with
Let L be an infinite horizontal plasma layer, permeated by an imposed uniform magnetic field \({\mathbf{H }}_0\) normal to the layer, under the action of a vertical gravity field \({\mathbf{g }}=-g\mathbf{k} \), and in which a constant adverse temperature gradient \(\beta \) is maintained. Let \(d>0\), \(\varOmega _d={{\mathbb {R}}}^2\times (0,d)\) and Oxyz be a Cartesian frame of reference with unit vectors \({\mathbf{i }}, {\mathbf{j }}, {\mathbf{k }}\), respectively. We assume that the plasma is confined between the planes \(z=0\) and \(z=d\), with assigned temperatures \({{{\tilde{T}}}}(x,y,0)={{{\tilde{T}}}}_0,\) \({{{\tilde{T}}}}(x,y,d)=-\beta d+{{{\tilde{T}}}}_0\). Here we consider the rest state \(m_0=(\tilde{\mathbf{v }},\tilde{{\mathbf{H }}}, {{{\tilde{T}}}}, {{{\tilde{p}}}})= (0,H_0{\mathbf{k }}, -\beta z+{{{\tilde{T}}}}_0, {{{\tilde{p}}}})\) (thermal conduction), which is a solution to the stationary previous equations. In the sequel, \({\mathbf{x }}=(x,y,z)\) will denote the coordinates in the unity of length d and, in view of the symmetry with respect to the two bounding planes, the origin of z is fixed at the midway between the two planes and one has then \(z\in \left[ -\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right] \). The following symbols are also used:
The (non-dimensional) equations for a perturbation \((\mathbf{u} ,{\mathbf{h }},\theta , p_1)\) to \(m_0\) are
To system (14), we add the kinematically admissible initial conditions
where \(\theta _0,{\mathbf{u }}_0,{\mathbf{h }}_0\) are assigned initial fields. As concerns the boundary conditions, we assume that the two planes bounding L are rigid and electricity perfectly conducting \(\{\)case B of Chandrasekhar (1981)\(\}\) and at assigned temperatures. Then in view of the fluid viscosity and its complete adherence to the rigid boundaries, there will be no motions on the boundaries:
On the other hand, in view of the perfect electric conductibility, no magnetic field can cross the boundary
Finally, since the boundaries are at assigned temperatures, one has
We assume that
-
(i)
the perturbations \((\nabla \pi ,u,v,w,\theta ,h_1,h_2,h)\) are periodic in the x and y directions of periods \(2\pi /a_x,\, 2\pi /a_y\), respectively;
-
(ii)
\(\varOmega =[0, 2\pi /a_x]\times [0,2\pi /a_y]\times \left[ -\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right] \) is the periodicity cell;
-
(iii)
\(u,v,w,\theta , h_1,h_2,h\) are such that together with all their first derivatives and second spatial derivatives are square integrable in \(\varOmega ,\forall t\in {{\mathbb {R}}}^+\) and can be expanded in a Fourier series uniformly convergent in \(\varOmega \).
We end by recalling that, along the time, many theorems of existence of weak and strong solution for MHD equations, under various circumstances, have been given. We confine ourselves to mentioning the recent papers concerning the existence of smooth solutions for large data \(\{\)see Lin et al. (2016), Ren et al. (2014), Zhou and Zhu (2018), Zhang (2019) and reference therein\(\}\).
3 Essential fields
Neglecting the non-linear terms of (14), one obtains
under the initial-boundary conditions analogous to (15) and (16). Since the vertical component of the double curl of (19)\(_1\) is
one has the linear system in the essential fields \(({{{\hat{w}}}}, {{{\hat{h}}}}, {{{\hat{\theta }}}})\) in \(\varOmega \times [0,\infty ]\)
with
which is the same class of perturbations considered in Chandrasekhar (1981) (normal modes). Therefore, setting
in view of the linearity of (21) one has
with
4 Additional boundary conditions
Requiring that (19)\(_2\) continues to hold also for \(z=\pm \displaystyle \frac{1}{2}\), the following supplementary boundary condition holds: \(\displaystyle \frac{\partial w}{\partial z}=0,\) at \(z=\pm \displaystyle \frac{1}{2}\).
In fact, (19)\(_2\) implies
Then
Moreover, in view of (16) and (18), (21)\(_2\) and (21)\(_3\) imply, respectively,
Collecting all the boundary conditions, one has
5 Admissible perturbations
One easily verifies that (29) is verified by (22), on choosing \(\forall z\in \left[ -\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right] \)
and \(F_{1n}\) solution of the Sturm–Liouville type problem
whose solutions are given by the set of the even orthogonal functions \(\{\)Chandrasekhar (1981), p. 635\(\}\)
and the set of the odd orthogonal functions
It is found that Chandrasekhar (1981)
in the case (32) and by
in the case (33).
6 Stability with \({{{\hat{w}}}}\) even
In view of
one has
and (24) implies
In the case \(F_{1n}=f_n\), the functions
are even in \(\left[ -\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right] \) and one has
Then (38) implies
i.e.
for any initial data and \(P_m, P_r, \sqrt{R}\) and \(\sqrt{Q}\). In view of (38)\(_2\) the exponential decay of \({{{\tilde{w}}}}\)—for any initial data—immediately follows. In Appendix, the linear and nonlinear \(L^2(\varOmega )\)-energy decay is investigated.
7 Stability with \({{{\hat{w}}}}\) odd
Setting
one has
Then (38) gives
with \(L_n\) given by
and
The spectrum equation of (42) is
with the characteristic values \({\texttt {I}}_{rn}, (r=1,2,3)\), given by \(\{\)see Appendix \(\}\)
8 Proof of property 1
The following properties hold.
-
(i)
Only if
$$\begin{aligned} {\texttt {I}}_{1n}<0,\,\,\,{\texttt {I}}_{2n}>0,\,\,\,{\texttt {I}}_{3n}<0,\,\,\,\forall (a^2,n)\in {{\mathbb {R}}}^+\times {{\mathbb {N}}}, \end{aligned}$$(47)the roots of (44) (eigenvalues of L) have all negative real parts (Appendix);
-
(ii)
if and only if the Liénard–Chipart conditions
$$\begin{aligned} {\texttt {I}}_{1n}<0,\,\,\,{\texttt {I}}_{2n}>0,\,\,\,{\texttt {I}}_{3n}<0,\,\,\, {\texttt {I}}_{1n}{\texttt {I}}_{2n}-{\texttt {I}}_{3n}<0,\,\, \forall n\in {{\mathbb {N}}} \end{aligned}$$(48)hold, all the eigenvalues have negative real part (Appendix 79).
Since \(\xi _n=a^2+4n^2\pi ^2\), \({\texttt {I}}_{1n}, {\texttt {I}}_{3n}\) are decreasing functions of n while \({\texttt {I}}_{2n}\) is an increasing function of n. Therefore, to satisfy i), it is necessary and sufficient to consider \(n=1\) (most destabilizing perturbation). Setting
-
1.
\({\texttt {I}}_1\) independent of Q and R given by
$$\begin{aligned} {\left\{ \begin{array}{ll} {\texttt {I}}_1=-\left( \displaystyle \frac{\gamma }{\xi }+\displaystyle \frac{P_m+P_r}{P_mP_r}\right) <0,\\ \\ \gamma =a^4+8a^2\pi ^2+\mu ^4; \end{array}\right. }\end{aligned}$$(50) -
2.
\({\texttt {I}}_2\) is a decreasing function of \(R, \forall (a^2, P_m, P_r, Q)\in ({{\mathbb {R}}}^+)^4\) and
$$\begin{aligned} {\texttt {I}}_2=0\Leftrightarrow {\left\{ \begin{array}{ll} R=R_2=\displaystyle \frac{F_2}{a^2},\\ F_2=\xi ^3+\left[ 4\pi ^2QP_r+\gamma (P_m+P_r)\right] \xi ; \end{array}\right. }\end{aligned}$$(51) -
3.
\({\texttt {I}}_3\) is an increasing function of \(R,\,\forall (a^2, P_m, P_r, Q)\in ({{\mathbb {R}}}^+)^4\) and
$$\begin{aligned} {\texttt {I}}_3=0\Leftrightarrow {\left\{ \begin{array}{ll} R=R_3=\displaystyle \frac{F_3}{a^2},\\ \\ F_3=(\gamma +4\pi ^2Q)\xi . \end{array}\right. } \end{aligned}$$(52)
Property 4
Let
Then at \((a^2, P_m, P_r, Q, R)\) an Hopf bifurcation occurs if and only if one has that
and has the frequency \(\omega ^2={\texttt {I}}_2=\displaystyle \frac{{\texttt {I}}_3}{{\texttt {I}}_1}\).
Proof
Let (4) admit the couple of pure imaginary roots \(\lambda =\pm i \omega \). Then one has
Vice versa, let (54) holds. Then (4) becomes
i.e.
having the roots \(\lambda _1={\texttt {I}}_1,\,\lambda _2=i\sqrt{{\texttt {I}}_2},\,\lambda _3=-i\sqrt{{\texttt {I}}_2}\). \(\square \)
Property 5
At \(R=0\), \(m_0\) is stable for any \(a^2\).
Proof
In fact, (42) reduces to
Being \(\gamma +4\pi ^2Q>0,\,\forall a^2\), the linear stability of \(m_0\) \(\forall a^2\), immediately follows. \(\square \)
Property 6
The critical numbers \(R_{c_2}, R_{c_3}\) are given by
Proof
In view of
it follows that exists a finite wave number \(a^2_{c_2}\), root of \(a^2\displaystyle \frac{\mathrm{{d}}F_2}{\mathrm{{d}}a^2}=F_2\), such that
Analogously one has that exists a finite wave number \(a^2_{c_3}\)—root of the equation \(a^2\displaystyle \frac{\mathrm{{d}}F_3}{\mathrm{{d}}a^2}=F_3\)—such that
\(\square \)
Property 7
Property 1 holds.
Proof
In view of properties 4 and 5, it remains only to show its sufficiency. Let then (6) holds. On the other hand, the stability of \(m_0\) at \(R=0\) for any wave number, implies—in view of the Liénard–Chipart conditions
But, at \(\{R=R_{c_2}, a^2=a^2_{c_2}\}\), one has
therefore,
implies the existence in \(]0, R_{c_2}[\) of an \( R_*\) such that
\(\square \)
9 Proof of Properties 2 and 3
Property 8
In plasma layers between two electricity perfectly conducting rigid planes, the MHD-Hopf bifurcations occur only in the plasma with \(P_m>P_r\) and only for Q sufficiently high. Precisely if and only if
Proof
(66) are necessary. In fact,
Let then \(P_r<P_m\) and set
In view of
one has
which implies \(R_{c_2}>R_{c_3}\). Therefore, \((\varPsi )_{(a^2=0)}<0\), i.e. (66)\(_2\) is necessary too. Vice versa, let (66) hold. Then
implies the existence of a positive number \(a^2_*\) such that
according to \(a^2\le (\ge ) a^2_*\), respectively, and one has
\(\square \)
Property 9
Property 3holds.
Proof
Property defines precisely the frontier between the onset of Hopf bifurcation (and hence the instability for \(R\ge R_{c_2}\)) and the onset of steady bifurcation (and hence the instability for \(R\ge R_{c_3}\)). Property 3 immediately follows. \(\square \)
Remark 1
We remark that \(a^2_*\) is obtained requiring \((\varPsi )_{(a^2=a^2_*)}=0\Leftrightarrow (F_2-F_3)_{(a^2=a^2_*)}=0\). One has
i.e.
Therefore,
and—in view of (66)\(_2\)—is guaranteed
and hence
Remark 2
In Fig. 1, the behaviour of \(Q_c\) at the increasing of \(P_m(>P_r)\) for \(P_r\in \{1,2,4\}\) is evaluated.
In Fig. 2, the behaviour of the ratio \({{{{\mathcal {R}}}}}\),
between the critical values of Q in the rigid–rigid case (given by (66)) and free–free case (given by (1)) at the growing of \(P_r\) is considered.
10 Hopf bifurcation threshold in the rigid–free and free–rigid cases in MHD
In view of
it follows that \(w_n,\theta _n,Z_n\) verify the conditions
One easily realizes that (74) and (75) are, respectively, the boundary MHD conditions for a layer of depth \(\displaystyle \frac{1}{2}\) in the free–rigid and rigid–free cases respectively, when the rigid plane bounding the layer is electricity perfectly conducting. Therefore, the results of the previous sections can be used but taking into account of the different depth. The Rayleigh and Chandrasekhar numbers of the free–rigid and rigid–free cases, \({{{\bar{R}}}}, {{{\bar{Q}}}}\) are defined by
which implies
11 Discussion, final remarks and perspectives
-
1.
The onset of Hopf bifurcations in plasma horizontal layers between two perfect conductor rigid planes and embedded in a constant transverse magnetic field, in non-relativistic thermal MHD, is investigated.
-
2.
The threshold of Hopf bifurcation characterization—via a simple formula—problem introduced by Chandrasekhar Chandrasekhar (1981)—is solved by introducing a critical value \(R_{c_2}\) for the Rayleigh number and requiring \(R_{c_2}<R_{c_3}\) with \(R_{c_3}\) steady convection critical value.
-
3.
The property of each coefficient of the spectrum equation to drive the instability and the type of bifurcation, is shown and applied.
-
4.
The threshold \(Q_c\) that, in the structural assumption \(P_r<P_m\), the Chandrasekhar number Q has to cross for the onset of Hopf bifurcations, is given by (66) in the rigid–rigid case and by (10.4) in the rigid–free and free–rigid cases.
-
5.
The onset of Hopf bifurcations in rotating layers between perfect conductor rigid planes, in MHD and in presence of Hall currents, is under investigation.
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Appendix
Appendix
1.1 Characteristic values of a \(3\times 3 \) matrix
Let \({{\mathcal {L}}}=\left\| a_{ij}\right\| \) be a quadratic matrix of order 3, with real entries. The algebraic cubic equation
with
is the spectral equation of \({{\mathcal {L}}}\) and its roots are the eigenvalues of \({{\mathcal {L}}}\). \(P(\lambda )\), being a polynomial of 3 degree, can be written (McLaurin)
In view of
setting
one has
and the spectral equation becomes
The quantities \({\texttt {I}}_1,{\texttt {I}}_2,{\texttt {I}}_3\) are called characteristic values or invariants of \({{\mathcal {L}}}\) since are invariant with respect to nonsingular linear transformations.
1.2 Liénard–Chipart criterion
Let \(a_{ij}\) be real numbers and let
with
Then (84) is the spectral equation of (85) i.e. the roots of (84) are the eigenvalues of (85). Therefore, the zero solution is stable if and only if all the eigenvalues have all negative real parts. The algebraic Liénard–Chipart’s criterion Merkin (1997); Gantmacher (2000), applied to (84), allows to determine the stability of the zero solution of (85) without solving the spectral equation. In fact, introducing the Hurwitz matrix of (84)
the criterion is the roots of (84) have negative real part if and only if
In particular, it follows that each one of the conditions
is necessary for all the roots have negative real parts (otherwise some roots will have positive real parts) (Merkin 1997; Gantmacher 2000).
1.3 \(L^2(\varOmega )\)-linear and nonlinear energy decay
Let \(R_L\) be the R critical value of linear stability of \(m_0\) and let \(\lambda _i,\, (i=1,2,3)\), be the eigenvalues associated to \(R<R_L\). Denoting by \(R_e(\lambda )\) the real part of an eigenvalue \(\lambda \) and setting
one has not only \({{\overline{\alpha }}}>0\) but also
with \(\hat{{{\mathcal {E}}}}\) partial \(L^2(\varOmega )\) linear energy given by
In fact, setting
(42) and (43), for \(n=1\), give
Since there exists a nonsingular transformation
such that
\(\{\)see Gantmacher (1975), p. 196, formula (10) with \(\delta =\displaystyle \frac{\alpha }{2},\eta =0\}\), then (95) and (96) imply
with \(|{\mathbf{X }}_0|\ge |U||{\mathbf{Y }}_0|\) and hence \(\forall |{\mathbf{X }}_0|\), since \(|{\mathbf{Y }}_0|\) is arbitrary. Setting
one has
and (91) immediately follows together with
Finally, introducing the total \(L^2(\varOmega )-\)linear energy
one easily obtains
and taking into account that the boundary conditions and the Poincaré inequality Flavin and Rionero (1996) imply
one easily obtains
and hence
In view of (100), one has
Passing to the nonlinear \(L^2(\varOmega )\)-energy decay and setting
in view of
one has
On the other hand, (103) and (106) imply
and hence
with \({{{\hat{E}}}}_0>2\sqrt{R_L} \hat{{{\mathcal {E}}}}_0\) and hence \(\forall {{{\hat{E}}}}_0\) since \(\hat{{{\mathcal {E}}}}_0\) is arbitrary. Setting
one has
On choosing
one has
i.e. \(R<R_L\) implies the global nonlinear asymptotic \(L^2(\varOmega )\)-stability.
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Rionero, S. Hopf bifurcations in plasma layers between rigid planes in thermal magnetohydrodynamics, via a simple formula. Rend. Fis. Acc. Lincei 31, 985–997 (2020). https://doi.org/10.1007/s12210-020-00955-1
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DOI: https://doi.org/10.1007/s12210-020-00955-1