×

A rigorous derivation and energetics of a wave equation with fractional damping. (English) Zbl 1483.35165

The authors consider the coupled system accounting for the motion of longitudinal elastic waves of a membrane coupled to viscous flows in the enclosing half-space. This system is written as \(\rho _{memb}\overset{..}{U} =\kappa \Delta _{x}U-\mu \partial _{z}v\), \(\overset{.}{U}(t,x)=v(t,x,0)\), for \(t>0\), \(x\in \Sigma \subset \mathbb{R}^{d-1}\), where \(U\) is the horizontal displacement of the membrane, \(\Sigma \) represents the membrane, and \(v\) is the horizontal velocity of the viscous fluid which satisfies \( \rho _{bulk}\overset{.}{v}=\mu \Delta _{x,z}v\), for \(t>0\), \((x,z)\in \Omega =\Sigma \times (-\infty ,0)\).The authors introduce the associated energy \( \mathbb{E}(U,\overset{.}{U},v)=\int_{\Sigma }(\frac{\rho _{memb}}{2}\overset{ ..}{U}^{2}+\frac{\kappa }{2}\left\vert \nabla U\right\vert ^{2})dx+\int_{\Omega }\frac{\rho _{bulk}}{2}v^{2}dzdx\) and the functional space \(\mathbf{H}=H^{1}(\Sigma )\times L^{2}(\Sigma )\times L^{2}(\Omega )\), and they prove that \(\mathbb{E}\) acts as a Lyapunov function and that it is a bounded quadratic form on \(\mathbf{H}\). They introduce a non-dimensional form of this system \(\overset{..}{U}=\Delta _{x}U-\partial _{z}v\mid _{z=0}\) , \(\overset{.}{U}=v\mid _{z=0}\), for \(t>0\), \(x\in \Sigma \), \(\overset{.}{v} =\varepsilon ^{2}\Delta _{x}v+\partial _{z}^{2}v\) for \(t>0\), \((x,z)\in \Omega \), with \(\varepsilon =\mu /\sqrt{\rho _{memb}k}\), and the operator \( A_{\varepsilon }:D(A_{\varepsilon })\subset \mathbf{H}\rightarrow \mathbf{H}\) through \(A_{\varepsilon }\left( \begin{array}{c} U \\ V \\ v \end{array} \right) =\left( \begin{array}{c} V \\ \Delta _{x}U-\partial _{z}v\mid _{z=0} \\ \varepsilon ^{2}\Delta _{x}v+\partial _{z}^{2}v \end{array} \right) \). The first main result proves that the operator \(A_{\varepsilon }\) is the generator of a strongly continuous semigroup. The second main result of the paper describes the asymptotic behavior of the solution to the original problem when \(\varepsilon \) goes to 0: for every initial condition \( w_{0}\in \mathbf{H}\), the semi-group solution \(w^{\varepsilon }(t)=e^{tA_{\varepsilon }}w_{0}\) strongly converges to some limit \(w^{0}\). In the last part of their paper, the authors prove that the damped wave equation \(\overset{..}{U}(t,x)+\int_{0}^{t}\frac{1}{\sqrt{\pi (t-\tau )}} \overset{..}{U}(\tau ,x)d\tau =\Delta U(t,x)\) on \(\Sigma \) carries a natural energy-dissipation structure.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74J15 Surface waves in solid mechanics
74K15 Membranes
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akagi, G., Fractional flows driven by subdifferentials in Hilbert spaces, Isr. J. Math., 234, 2, 809-862 (2019) · Zbl 1431.35221 · doi:10.1007/s11856-019-1936-9
[2] Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqns., 32, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306
[3] El Hady, A.; Machta, BB, Mechanical surface waves accompany action potential propagation, Nature Commun., 6, 6697 (2015) · doi:10.1038/ncomms7697
[4] Escher, J., Nonlinear elliptic systems with dynamic boundary conditions, Math. Z., 210, 413-439 (1992) · Zbl 0759.35025 · doi:10.1007/BF02571805
[5] Griesbauer, J.; Bössinger, S.; Wixforth, A.; Schneider, MF, Propagation of 2D pressure pulses in lipid monolayers and its possible implications for biology, Phys. Rev. Lett., 108, 198103 (2012) · doi:10.1103/PhysRevLett.108.198103
[6] G. Gripenberg, S.-O. Londen, and O. Staffans. Volterra Integral and Functional Equations. Cambridge University Press, 1990. · Zbl 0695.45002
[7] Galé, JE; Miana, PJ; Stinga, PR, Extension problem and fractional operators: semigroups and wave equations, J. Evol. Eqns., 13, 343-368 (2013) · Zbl 1336.47049 · doi:10.1007/s00028-013-0182-6
[8] Gómez-Aguilar, JF; Yépez-Martínez, H.; Calderón-Ramón, C.; Cruz-Orduña, I.; Escobar-Jiménez, RF; Olivares-Peregrino, VH, Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel, Entropy, 17, 6289-6303 (2015) · Zbl 1338.70026 · doi:10.3390/e17096289
[9] J. Kappler and R. R. Netz. Multiple surface wave solution on linear viscoelastic media. Europhys. Lett., 112(1), 19002/6 pp., 2015.
[10] Kim, GH; Kosterin, P.; Obaid, AL; Salzberg, BM, A mechanical spike accompanies the action potential in mammalian nerve terminals, Biophysical Journal, 92, 9, 3122-3129 (2007) · doi:10.1529/biophysj.106.103754
[11] J. Kemppainen, J. Siljander, V. Vergara, and R. Zacher. Decay estimates for time-fractional and other non-local in time subdiffusion equations in \({\mathbb{R}}^d\). Math. Ann., 366(3-4), 941-979, 2016. · Zbl 1354.35178
[12] J. Kappler, S. Shrivastava, M. F. Schneider, and R. R. Netz. Nonlinear fractional waves at elastic interfaces. Phys. Rev. Fluids, 2(11), 114804, 2017. (look for supplement).
[13] J. Kappler, S. Shrivastava, M. F. Schneider, and R. R. Netz. Nonlinear fractional waves at elastic interfaces – supplemental information. Phys. Rev. Fluids, 2(11), 114804/suppl. 26 pp., 2017.
[14] Lions, J-L; Magenes, E., Non-homogeneous boundary value problems and applications (1972), New York: Springer-Verlag, New York · Zbl 0227.35001 · doi:10.1007/978-3-642-65217-2
[15] J. Lucassen. Longitudinal capillary waves. Part 1. Theory. Trans. Faraday Soc., 64, 2221-2229, 1968.
[16] Meyer, HH, Zur Theorie der Alkoholnarkose, Arch. Exp. Pathol. Pharmakol., 42, 2-4, 109-118 (1899) · doi:10.1007/BF01834479
[17] Oldham, KB, Fractional differential equations in electrochemistry, Adv. Engin. Softw., 41, 9-12 (2010) · Zbl 1177.78041 · doi:10.1016/j.advengsoft.2008.12.012
[18] Overton, CE, Studien über die Narkose zugleich ein Beitrag zur allgemeinen Pharmakologie (1901), Jena, Germany: Gustav Fischer, Jena, Germany
[19] A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer New York, 1983. · Zbl 0516.47023
[20] Prüss, J.; Vergara, V.; Zacher, R., Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory, Discr. Cont. Dynam. Systems, 26, 2, 625-647 (2010) · Zbl 1191.45005 · doi:10.3934/dcds.2010.26.625
[21] Shrivastava, S.; Schneider, MF, Evidence for two-dimensional solitary sound waves in a lipid controlled interface and its implications for biological signalling, J. Royal Soc. Interface, 11, 20140098 (2014) · doi:10.1098/rsif.2014.0098
[22] Tasaki, I.; Byrne, PM, Heat production associated with a propagated impulse in bullfrog myelinated nerve fibers, Japan. J. Physiol., 42, 805-813 (1992) · doi:10.2170/jjphysiol.42.805
[23] Vergara, V.; Zacher, R., Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z., 259, 2, 287-309 (2008) · Zbl 1144.45003 · doi:10.1007/s00209-007-0225-1
[24] Vergara, V.; Zacher, R., A priori bounds for degenerate and singular evolutionary partial integro-differential equations, Nonlinear Anal., 73, 11, 3572-3585 (2010) · Zbl 1205.45011 · doi:10.1016/j.na.2010.07.039
[25] Vergara, V.; Zacher, R., Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Analysis, 47, 1, 210-239 (2015) · Zbl 1317.45006 · doi:10.1137/130941900
[26] Vergara, V.; Zacher, R., Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations, J. Evol. Equ., 17, 1, 599-626 (2017) · Zbl 1365.35218 · doi:10.1007/s00028-016-0370-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.