Local weak solvability of a moving boundary problem describing swelling along a halfline

Previous Article
On the local and global existence of solutions to 1d transport equations with nonlocal velocity
NHM Home
This Issue
Next Article

September 2019, 14(3): 445-469. doi: 10.3934/nhm.2019018

Local weak solvability of a moving boundary problem describing swelling along a halfline

Kota Kumazaki ^1, and Adrian Muntean ^2,

1.	Nagasaki University, Department of Education, 1-14, Bunkyo-cho, Nagasaki, 852-8521, Japan
2.	Karlstad University, Department of Mathematics and Computer Science, Universitetsgatan 2,651 88 Karlstad, Sweden

Received May 2018 Revised February 2019 Published May 2019

We obtain the local well-posedness of a moving boundary problem that describes the swelling of a pocket of water within an infinitely thin elongated pore (i.e. on $ [a, +\infty), \ a>0 $). Our result involves fine a priori estimates of the moving boundary evolution, Banach fixed point arguments as well as an application of the general theory of evolution equations governed by subdifferentials.

Keywords: Moving boundary problem, a priori estimates, Nonlinear initial-boundary value problems for nonlinear parabolic equations, Swelling of pores, Flux boundary conditions.

Mathematics Subject Classification: Primary: 35R35; Secondary: 35B45, 35K61.

Citation: Kota Kumazaki, Adrian Muntean. Local weak solvability of a moving boundary problem describing swelling along a halfline. Networks and Heterogeneous Media, 2019, 14 (3) : 445-469. doi: 10.3934/nhm.2019018

References:

[1]	T. Aiki, Y. Murase, N. Sato and K. Shirakawa, A mathematical model for a hysteresis appearing in adsorption phenomena, SūrikaisekikenkyūshoKōkyūroku, 1856 (2013), 1-12.
[2]	T. Aiki and Y. Murase, On a large time behavior of a solution to a one-dimensional free boundary problem for adsorption phenomena, J. Math. Anal. Appl., 445 (2017), 837-854. doi: 10.1016/j.jmaa.2016.06.008.
[3]	T. Aiki and A. Muntean, Existence and uniqueness of solutions to a mathematical model predicting service life of concrete structures, Adv. Math. Sci. Appl., 19 (2009), 109-129.
[4]	T. Aiki and A. Muntean, Large time behavior of solutions to a moving-interface problem modeling concrete carbonation, Comm. Pure Appl. Anal., 9 (2010), 1117-1129. doi: 10.3934/cpaa.2010.9.1117.
[5]	T. Aiki and A. Muntean, A free-boundary problem for concrete carbonation: Rigorous justification of $\sqrt{t}$–law of propagation, Interface. Free Bound., 15 (2013), 167-180. doi: 10.4171/IFB/299.
[6]	A. Fasano, G. Meyer and M. Primicerio, On a problem in the polymer industry: Theoretical and numerical investigation of swelling, SIAM J. Appl. Math., 17 (1986), 945-960. doi: 10.1137/0517067.
[7]	A. Fasano and A. Mikelic, The 3D flow of a liquid through a porous medium with adsorbing and swelling granules, Interface. Free Bound., 4 (2002), 239-261. doi: 10.4171/IFB/60.
[8]	T. Fatima, A. Muntean and T. Aiki, Distributed space scales in a semilinear reaction-diffusion system including a parabolic variational inequality: A well-posedness study, Adv. Math. Sci. Appl., 22 (2012), 295-318.
[9]	B. W. van de Fliert and R. van der Hout, A generalized Stefan problem in a diffusion model with evaporation, SIAM J. Appl. Math., 60 (2000), 1128-1136. doi: 10.1137/S0036139997327599.
[10]	A. Friedman and A. Tzavaras, A quasilinear parabolic system arising in modelling of catalytic reactors, J. Differential Equations, 70 (1987), 167-196. doi: 10.1016/0022-0396(87)90162-8.
[11]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.
[12]	K. Kumazaki, T. Aiki, N. Sato and Y. Murase, Multiscale model for moisture transport with adsorption phenomenon in concrete materials, Appl. Anal., 97 (2018), 41-54. doi: 10.1080/00036811.2017.1325473.
[13]	K. Kumazaki and A. Muntean, Global weak solvability of a moving boundary problem describing swelling along a halfline, arXiv: 1810.08000.
[14]	A. Muntean and M. Böhm, A moving boundary problem for concrete carbonation: global existence and uniqueness of solutions, J. Math. Anal. Appl., 350 (2009), 234-251. doi: 10.1016/j.jmaa.2008.09.044.
[15]	A. Muntean and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media, J. Math. Anal. Appl., 37 (2010), 705-718. doi: 10.1016/j.jmaa.2010.05.056.
[16]	T. L. van Noorden and I. S. Pop, A Stefan problem modelling crystal dissolution and precipitation, IMA J. Appl. Math., 73 (2008), 393-411. doi: 10.1093/imamat/hxm060.
[17]	T. L. van Noorden, I. S. Pop, A. Ebigbo and R. Helmig, An upscaled model for biofilm growth in a thin strip, Water Resour. Res., 46 (2010), 1-14. doi: 10.1029/2009WR008217.
[18]	N. Sato, T. Aiki, Y. Murase and K. Shirakawa, A one dimensional free boundary problem for adsorption phenomena, Netw. Heterog. Media, 9 (2014), 655-668. doi: 10.3934/nhm.2014.9.655.
[19]	M. J. Setzer, Micro-ice-lens formation in porous solid, J. Colloid Interface Sci., 243 (2001), 193-201. doi: 10.1006/jcis.2001.7828.
[20]	X. Weiqing, The Stefan problem with a kinetic condition at the free boundary, SIAM J. Math. Anal., 21 (1990), 362-373. doi: 10.1137/0521020.
[21]	M. Zaal, Cell swelling by osmosis: A variational approach, Interface. Free Bound., 14 (2012), 487-520. doi: 10.4171/IFB/289.

show all references

References:

[1]	T. Aiki, Y. Murase, N. Sato and K. Shirakawa, A mathematical model for a hysteresis appearing in adsorption phenomena, SūrikaisekikenkyūshoKōkyūroku, 1856 (2013), 1-12.
[2]	T. Aiki and Y. Murase, On a large time behavior of a solution to a one-dimensional free boundary problem for adsorption phenomena, J. Math. Anal. Appl., 445 (2017), 837-854. doi: 10.1016/j.jmaa.2016.06.008.
[3]	T. Aiki and A. Muntean, Existence and uniqueness of solutions to a mathematical model predicting service life of concrete structures, Adv. Math. Sci. Appl., 19 (2009), 109-129.
[4]	T. Aiki and A. Muntean, Large time behavior of solutions to a moving-interface problem modeling concrete carbonation, Comm. Pure Appl. Anal., 9 (2010), 1117-1129. doi: 10.3934/cpaa.2010.9.1117.
[5]	T. Aiki and A. Muntean, A free-boundary problem for concrete carbonation: Rigorous justification of $\sqrt{t}$–law of propagation, Interface. Free Bound., 15 (2013), 167-180. doi: 10.4171/IFB/299.
[6]	A. Fasano, G. Meyer and M. Primicerio, On a problem in the polymer industry: Theoretical and numerical investigation of swelling, SIAM J. Appl. Math., 17 (1986), 945-960. doi: 10.1137/0517067.
[7]	A. Fasano and A. Mikelic, The 3D flow of a liquid through a porous medium with adsorbing and swelling granules, Interface. Free Bound., 4 (2002), 239-261. doi: 10.4171/IFB/60.
[8]	T. Fatima, A. Muntean and T. Aiki, Distributed space scales in a semilinear reaction-diffusion system including a parabolic variational inequality: A well-posedness study, Adv. Math. Sci. Appl., 22 (2012), 295-318.
[9]	B. W. van de Fliert and R. van der Hout, A generalized Stefan problem in a diffusion model with evaporation, SIAM J. Appl. Math., 60 (2000), 1128-1136. doi: 10.1137/S0036139997327599.
[10]	A. Friedman and A. Tzavaras, A quasilinear parabolic system arising in modelling of catalytic reactors, J. Differential Equations, 70 (1987), 167-196. doi: 10.1016/0022-0396(87)90162-8.
[11]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.
[12]	K. Kumazaki, T. Aiki, N. Sato and Y. Murase, Multiscale model for moisture transport with adsorption phenomenon in concrete materials, Appl. Anal., 97 (2018), 41-54. doi: 10.1080/00036811.2017.1325473.
[13]	K. Kumazaki and A. Muntean, Global weak solvability of a moving boundary problem describing swelling along a halfline, arXiv: 1810.08000.
[14]	A. Muntean and M. Böhm, A moving boundary problem for concrete carbonation: global existence and uniqueness of solutions, J. Math. Anal. Appl., 350 (2009), 234-251. doi: 10.1016/j.jmaa.2008.09.044.
[15]	A. Muntean and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media, J. Math. Anal. Appl., 37 (2010), 705-718. doi: 10.1016/j.jmaa.2010.05.056.
[16]	T. L. van Noorden and I. S. Pop, A Stefan problem modelling crystal dissolution and precipitation, IMA J. Appl. Math., 73 (2008), 393-411. doi: 10.1093/imamat/hxm060.
[17]	T. L. van Noorden, I. S. Pop, A. Ebigbo and R. Helmig, An upscaled model for biofilm growth in a thin strip, Water Resour. Res., 46 (2010), 1-14. doi: 10.1029/2009WR008217.
[18]	N. Sato, T. Aiki, Y. Murase and K. Shirakawa, A one dimensional free boundary problem for adsorption phenomena, Netw. Heterog. Media, 9 (2014), 655-668. doi: 10.3934/nhm.2014.9.655.
[19]	M. J. Setzer, Micro-ice-lens formation in porous solid, J. Colloid Interface Sci., 243 (2001), 193-201. doi: 10.1006/jcis.2001.7828.
[20]	X. Weiqing, The Stefan problem with a kinetic condition at the free boundary, SIAM J. Math. Anal., 21 (1990), 362-373. doi: 10.1137/0521020.
[21]	M. Zaal, Cell swelling by osmosis: A variational approach, Interface. Free Bound., 14 (2012), 487-520. doi: 10.4171/IFB/289.

[1]	Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244
[2]	Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205
[3]	Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[4]	Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637
[5]	Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212
[6]	Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917
[7]	Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[8]	Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487
[9]	Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[10]	Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
[11]	Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $ x $-dependent coefficients. Evolution Equations and Control Theory, 2020, 9 (1) : 153-179. doi: 10.3934/eect.2020001
[12]	V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191
[13]	Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure and Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[14]	Rusuo Ye, Yi Zhang. Initial-boundary value problems for the two-component complex modified Korteweg-de Vries equation on the interval. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022111
[15]	Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
[16]	Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[17]	Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[18]	Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381
[19]	Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[20]	Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015

2020 Impact Factor: 1.213

Tools

Metrics

Other articles
by authors

on AIMS
Kota Kumazaki Adrian Muntean
on Google Scholar
Kota Kumazaki Adrian Muntean

2020 Impact Factor: 1.213

Tools

Article outline

2020 Impact Factor: 1.213

Tools

Metrics

Other articles
by authors

on AIMS
Kota Kumazaki Adrian Muntean
on Google Scholar
Kota Kumazaki Adrian Muntean

[Back to Top]