| 1. |
- Aiki, Toyohiko, et al.
(author)
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A free boundary problem describing migration into rubbers : Quest of the large time behavior
- 2022
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In: Zeitschrift für angewandte Mathematik und Mechanik. - : John Wiley & Sons. - 0044-2267 .- 1521-4001. ; 102:7
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Journal article (peer-reviewed)abstract
- In many industrial applications, rubber-based materials are routinely used in conjunction with various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theoretically the penetration depth as well as the amount of diffusants stored inside the material. In this framework, we prove the global solvability and explore the large time-behavior of solutions to a one-phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber. The key idea in the proof of the large time behavior is to benefit from a contradiction argument, since it is difficult to obtain uniform estimates for the growth rate of the free boundary due to the use of a Robin boundary condition posed at the fixed boundary.
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| 2. |
- Aiki, Toyohiko, et al.
(author)
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Differentiability of a solution of a free boundary problem describing water adsorption
- 2020
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In: Advances in Mathematical Sciences and Applications. - : GAKKO TOSHO CO. - 1343-4373. ; 29, s. 247-282
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Journal article (peer-reviewed)abstract
- Sato-Aiki-Murase-Shirakawa [7, 16] proposed a free boundary problem in order to describe water adsorption appearing in moisture transport of concrete carbonation process. The problem is to find a pair of a curve and a function for given boundary and initial functions. In this paper we consider that the boundary and initial functions depend on some parameter. Kumazaki [9, 10] already proved the continuous and measurable properties of solutions with respect to the parameter. The aim of this paper is to establish differentiability of solutions with respect to the parameter as a new property. This result will be applied to research of the multi-scale model to water adsorption in concrete carbonation.
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| 3. |
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| 4. |
- Kumazaki, Kota, et al.
(author)
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Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces
- 2020
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In: Interfaces and free boundaries (Print). - : European Mathematical Society Publishing House. - 1463-9963 .- 1463-9971. ; 22:1, s. 27-49
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Journal article (peer-reviewed)abstract
- We prove a global existence result for weak solutions to a free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a priori unknown interface never disappears. As main ingredients of the global existence proof, we rely on a local weak solvability result for our problem (as reported in [7]), uniform energy estimates of the solution, integral estimates on quantities defined at the free boundary, as well as a fine pointwise lower bound for the position of the moving boundary. Some of the estimates are time independent. They allow us to explore the large-time behavior of the position of the moving boundary. The approach is specific to one-dimensional settings.
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| 5. |
- Kumazaki, Kota, et al.
(author)
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Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect
- 2023
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In: Networks and Heterogeneous Media. - : American Institute of Mathematical Sciences. - 1556-1801 .- 1556-181X. ; 18:1, s. 80-108
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Journal article (peer-reviewed)abstract
- We consider a one-dimensional free boundary problem describing the migration of diffusants into rubber. In our setting, the free boundary represents the position of the front delimitating the diffusant region. The growth rate of this region is described by an ordinary differential equation that includes the effect of breaking the growth of the diffusant region. In this specific context, the breaking mechanism is should be perceived as a non-dissipative way of describing eventual hyperelastic response to a too fast diffusion penetration. In recent works, we considered a similar class of free boundary problems modeling diffusants penetration in rubbers, but without attempting to deal with the possibility of breaking or accelerating the occurring free boundaries. For simplified settings, we were able to show the global existence and uniqueness as well as the large time behavior of the corresponding solutions to our formulations. Since here the breaking effect is contained in the free boundary condition, our previous results are not anymore applicable. The main mathematical obstacle in ensuring the existence of a solution is the non-monotonic structure of the free boundary. In this paper, we establish the existence and uniqueness of a weak solution to the free boundary problem with breaking effect and give explicitly the maximum value that the free boundary can reach.
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| 6. |
- Kumazaki, Kota, et al.
(author)
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Local weak solvability of a moving boundary problem describing swelling along a halfline
- 2019
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In: Networks and Heterogeneous Media. - : American Institute of Mathematical Sciences. - 1556-1801 .- 1556-181X. ; 14:3, s. 445-469
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Journal article (peer-reviewed)abstract
- 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, +∞), 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.
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