Parabolic quasi-variational diffusion problems with gradient constraints
Nobuyuki Kenmochi
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 423-438 doi: 10.3934/dcdss.2013.6.423
In this paper we consider some mechanical phenomena whose dynamics is described by a class of quasi-variational inequalities of parabolic type. Our system consists of a second-order parabolic variational inequality with gradient constraint depending on the temperature and the heat equation. Since the temperature is unknown in our problem, the constraint function is unknown as well. In this sense, our problem includes the quasi-variational structure, and in the mathamtical analysis one of main difficulties comes from it. Our approach to the problem is based on the abstract theory of quasi-variational inequalities with non-local constraint which has been developed in [6]. However the abstract theory is not directly used in the existence proof of a solution, since the mathematical situation of the problem is much nicer than that in the abstract theory [6]. In this paper we prove the existence of a weak solution of our system.
keywords: superconductivity gradient constraint. subdifferentials Quasi-variational inequality
Phase-field systems with vectorial order parameters including diffusional hysteresis effects
Nobuyuki Kenmochi Jürgen Sprekels
Communications on Pure & Applied Analysis 2002, 1(4): 495-511 doi: 10.3934/cpaa.2002.1.495
This paper is concerned with phase-field systems of Penrose-Fife type which model the dynamics of a phase transition with non-conserved vectorial order parameter. The main novelty of the model is that the evolution of the order parameter vector is governed by a system consisting of one partial differential equation and one partial differential inclusion, which in the simplest case may be viewed as a diffusive approximation of the so-called multi-dimensional stop operator, which is one of the fundamental hysteresis operators. Results concerning existence, uniqueness and continuous dependence on data are presented which can be viewed as generalizations of recent results by the authors to cases where a diffusive hysteresis occurs.
keywords: Parabolic systems phase transitions. hysteresis phase-field models a priori estimates existence uniqueness
Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint
Nobuyuki Kenmochi Noriaki Yamazaki
Conference Publications 2011, 2011(Special): 824-833 doi: 10.3934/proc.2011.2011.824
We consider a phase-fi eld model of grain boundary motion with constraint, which is a nonlinear system of Kobayashi-Warren-Carter type: a nonlinear parabolic partial diff erential equation and a nonlinear parabolic variational inequality. Recently the existence of solutions to our system was shown in the N-dimensional case. Also the uniqueness was proved in the case when the space dimensional is one and initial data are good. In this paper we study the asymptotic stability of our model without uniqueness. In fact we shall construct global attractors for multivalued semigroups (multivalued semiflows) associated with our system in the N-dimensional case.
keywords: singular di usion equation grain boundary motion Global attractor multivalued semigroups
Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences
Toyohiko Aiki Joost Hulshof Nobuyuki Kenmochi Adrian Muntean
Discrete & Continuous Dynamical Systems - S 2014, 7(1): i-iii doi: 10.3934/dcdss.2014.7.1i
The more one dives into the structural details of material or life sciences problems, the more sophisticated and specific the mathematical tools needed to address these problems become. The challenges are generally twofold: On the one hand one wishes to find accurate descriptions of the microscale, while on the other hand, having in view certain microscale dynamics (close to micro phase transitions), one wishes to capture a basic understanding over much larger scales. In both cases one aims at well-posed PDE models in the sense of Hadamard, which are not only computable numerically but also verifiable against experiments.

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Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$
A. Damlamian Nobuyuki Kenmochi
Discrete & Continuous Dynamical Systems - A 1999, 5(2): 269-278 doi: 10.3934/dcds.1999.5.269
This paper is concerned with the subdifferential operator approach to nonlinear (possibly degenerate and singular) parabolic PDE's of the form $u_t-\Delta \beta(u) \ni f$ formulated in the dual space of $H^1(\Omega)$, where $\beta$ is a maximal monotone graph in $\mathbf R\times \mathbf R$. In the set-up considered so far [8], some coerciveness condition has been required for $\beta$, corresponding at least to the fact that it is onto $\mathbf R$. In the present paper, we show that the subdifferential operator approach is possible for any maximal monotone graph $\beta$ without any growth condition.
keywords: subdifferential Nonlinear semigroups degenerate Stefan problem.
Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint
Takeshi Fukao Nobuyuki Kenmochi
Discrete & Continuous Dynamical Systems - A 2015, 35(6): 2523-2538 doi: 10.3934/dcds.2015.35.2523
This paper is concerned with a heat convection problem. We discuss it in the framework of parabolic variational inequalities. The problem is a system of a heat equation with convection and a Navier-Stokes variational inequality with temperature-dependent velocity constraint. Our problem is a sort of parabolic quasi-variational inequalities in the sense that the constraint set for the velocity field depends on the unknown temperature. We shall give an existence result of the heat convection problem in a weak sense, and show that under some additional constraint for temperature there exists a strong solution of the problem.
keywords: Quasi-variational inequality velocity constraint Navier-Stokes equation evolution equation. heat convection
A thermohydraulics model with temperature dependent constraint on velocity fields
Takeshi Fukao Nobuyuki Kenmochi
Discrete & Continuous Dynamical Systems - S 2014, 7(1): 17-34 doi: 10.3934/dcdss.2014.7.17
In this paper, the Navier-Stokes variational inequality with the temperature dependent constraint is considered in 3-dimensional space. This problem is motivated by an initial-boundary value problem for a thermohydraulics model in which the absolute value of the velocity field is constrained, depending on the unknown temperature. The abstract theory of nonlinear evolution equations governed by subdifferentials of time-dependent convex functionals is useful in showing the existence of a solution. In the mathematical treatment, the point of emphasis is to specify a class of time-dependence of convex constraints.
keywords: Lagrange multiplier subdifferential volume preservation. Parabolic variational inequality time-dependent constraint
Abstract theory of variational inequalities and Lagrange multipliers
Takeshi Fukao Nobuyuki Kenmochi
Conference Publications 2013, 2013(special): 237-246 doi: 10.3934/proc.2013.2013.237
In this paper, the existence and uniqueness questions of abstract parabolic variational inequalities are considered in connection with Lagrange multipliers. The focus of authors' attention is the characterization of parabolic variational inequalities by means of Lagrange multipliers. It is well-known that various kinds of parabolic differential equations under convex constraints are represented by variational inequalities with time-dependent constraints, and the usage of Lagrange multipliers associated with constraints makes it possible to reformulate the variational inequalities as equations. In this paper, as a typical case, a parabolic problem with nonlocal time-dependent obstacle is treated in the framework of abstract evolution equations governed by time-dependent subdifferentials.
keywords: Lagrange multiplier. Parabolic variational inequality subdifferential
Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces
Akio Ito Noriaki Yamazaki Nobuyuki Kenmochi
Conference Publications 1998, 1998(Special): 327-349 doi: 10.3934/proc.1998.1998.327
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Elliptic Quasi-variational inequalities and applications
Yusuke Murase Risei Kano Nobuyuki Kenmochi
Conference Publications 2009, 2009(Special): 583-591 doi: 10.3934/proc.2009.2009.583
It is the main objective of this paper to discuss applications of the abstract results recently evolved in [16,17] to the solvability of variational inequalities with constraints depending on the unknown functions, which are called quasi-variational inequalities.
keywords: Elliptic quasi-variational inequality semi-monotone

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