- #1
somasimple
Gold Member
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Hello all,
I am working on a coupled hydraulic-electric model arising in nerve-membrane biophysics, in which the dynamics takes the singularly perturbed canonical form
$$
\partial_t x = f(x, y; \partial_\xi), \qquad \varepsilon, \partial_t y = g(x, y),
$$
with slow variable ##x = (p, q) \in \mathbb{R}^2## (a hydraulic pressure-flux pair obeying a Moens-Korteweg wave equation) and fast variable ##y = (\theta, V) \in \mathbb{R}^2## (a Langmuir adsorption fraction and an electrical potential). The small parameter is ##\varepsilon = \tau_{\mathrm{el}} / \tau_{\mathrm{hyd}}## and ranges from ##10^{-2}## to ##10^{-4}## in the physiological regime. The algebraic constraint ##g(x,y) = 0## takes an explicit Donnan-Langmuir form for which the Jacobian ##\partial_y g## at the critical manifold has eigenvalues with magnitudes ##|\lambda_\theta| \sim 1/\tau_{\mathrm{ion}}## and ##|\lambda_V| \sim 1/\tau_{\mathrm{el}}##, both real and negative on the admissible domain.
I have a pointwise Fenichel-Tikhonov persistence theorem and a constructive expansion ##h(x,\varepsilon) = h_0(x) + \varepsilon h_1(x) + O(\varepsilon^2)##. The full setup, the proof, and the explicit Donnan-Langmuir form of ##g## are in the attached PDF (6 pages).
I would like mathematical scrutiny on three points:
1. Verification of (R1)-(R3) under the explicit form of ##g##. Is the Donnan-Langmuir form sufficient, or do we need additional hypotheses?
1. Soundness of the constructive expansion. The recursion is solvable formally; is the resulting series convergent on a uniform ##\varepsilon##-neighbourhood, or only Borel-summable?
1. Bridge to the spatially extended setting. The slow operator carries hyperbolic spatial derivatives. Two strategies suggest themselves: (a) reduction along Moens-Korteweg characteristics of the slow operator, parametrising the slow-fast decomposition by the characteristic coordinate; or (b) direct application of the Bates-Lu-Zeng infinite-dimensional Fenichel theorem [1]. Is the characteristic argument rigorous as stated, or does it require an additional commutation hypothesis between the slow operator and the projection onto ##M_0##? Equivalently, which of (a) and (b) is cleaner for this specific system?
Comments and corrections are very welcome.
[1] Bates, Lu, Zeng. Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space.
I am working on a coupled hydraulic-electric model arising in nerve-membrane biophysics, in which the dynamics takes the singularly perturbed canonical form
$$
\partial_t x = f(x, y; \partial_\xi), \qquad \varepsilon, \partial_t y = g(x, y),
$$
with slow variable ##x = (p, q) \in \mathbb{R}^2## (a hydraulic pressure-flux pair obeying a Moens-Korteweg wave equation) and fast variable ##y = (\theta, V) \in \mathbb{R}^2## (a Langmuir adsorption fraction and an electrical potential). The small parameter is ##\varepsilon = \tau_{\mathrm{el}} / \tau_{\mathrm{hyd}}## and ranges from ##10^{-2}## to ##10^{-4}## in the physiological regime. The algebraic constraint ##g(x,y) = 0## takes an explicit Donnan-Langmuir form for which the Jacobian ##\partial_y g## at the critical manifold has eigenvalues with magnitudes ##|\lambda_\theta| \sim 1/\tau_{\mathrm{ion}}## and ##|\lambda_V| \sim 1/\tau_{\mathrm{el}}##, both real and negative on the admissible domain.
I have a pointwise Fenichel-Tikhonov persistence theorem and a constructive expansion ##h(x,\varepsilon) = h_0(x) + \varepsilon h_1(x) + O(\varepsilon^2)##. The full setup, the proof, and the explicit Donnan-Langmuir form of ##g## are in the attached PDF (6 pages).
I would like mathematical scrutiny on three points:
1. Verification of (R1)-(R3) under the explicit form of ##g##. Is the Donnan-Langmuir form sufficient, or do we need additional hypotheses?
1. Soundness of the constructive expansion. The recursion is solvable formally; is the resulting series convergent on a uniform ##\varepsilon##-neighbourhood, or only Borel-summable?
1. Bridge to the spatially extended setting. The slow operator carries hyperbolic spatial derivatives. Two strategies suggest themselves: (a) reduction along Moens-Korteweg characteristics of the slow operator, parametrising the slow-fast decomposition by the characteristic coordinate; or (b) direct application of the Bates-Lu-Zeng infinite-dimensional Fenichel theorem [1]. Is the characteristic argument rigorous as stated, or does it require an additional commutation hypothesis between the slow operator and the projection onto ##M_0##? Equivalently, which of (a) and (b) is cleaner for this specific system?
Comments and corrections are very welcome.
[1] Bates, Lu, Zeng. Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space.