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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 & 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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.   Google Scholar

[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.  Google Scholar

[13]

K. Kumazaki and A. Muntean, Global weak solvability of a moving boundary problem describing swelling along a halfline, arXiv: 1810.08000. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. Zaal, Cell swelling by osmosis: A variational approach, Interface. Free Bound., 14 (2012), 487-520.  doi: 10.4171/IFB/289.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.   Google Scholar

[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.  Google Scholar

[13]

K. Kumazaki and A. Muntean, Global weak solvability of a moving boundary problem describing swelling along a halfline, arXiv: 1810.08000. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. Zaal, Cell swelling by osmosis: A variational approach, Interface. Free Bound., 14 (2012), 487-520.  doi: 10.4171/IFB/289.  Google Scholar

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