# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

February 2014 , Volume 7 , Issue 1

Issue on analysis of non-equilibrium evolution problems: Selected topics in material and life sciences

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2014, 7(1): i-iii doi: 10.3934/dcdss.2014.7.1i +[Abstract](1555) +[PDF](110.7KB)
Abstract:
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.

2014, 7(1): 1-16 doi: 10.3934/dcdss.2014.7.1 +[Abstract](2391) +[PDF](422.0KB)
Abstract:
This paper is concerned with doubly nonlinear parabolic equations involving variable exponents. The existence of solutions is proved by developing an abstract theory on doubly nonlinear evolution equations governed by gradient operators. In contrast to constant exponent cases, two nonlinear terms have inhomogeneous growth and some difficulty may occur in establishing energy estimates. Our method of proof relies on an efficient use of Legendre-Fenchel transforms of convex functionals and an energy method.
2014, 7(1): 17-34 doi: 10.3934/dcdss.2014.7.17 +[Abstract](1618) +[PDF](417.8KB)
Abstract:
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.
2014, 7(1): 35-51 doi: 10.3934/dcdss.2014.7.35 +[Abstract](1416) +[PDF](470.3KB)
Abstract:
An equation for the dynamics of the vesicle supply center model of tip growth in fungal hyphae is derived. For this we analytically prove the existence and uniqueness of a traveling wave solution which exhibits the experimentally observed behavior. The linearized dynamics around this solution is analyzed and we conclude that all eigenmodes decay in time. Numerical calculation of the first eigenvalue gives a timescale in which small perturbations will die out.
2014, 7(1): 53-62 doi: 10.3934/dcdss.2014.7.53 +[Abstract](1518) +[PDF](343.7KB)
Abstract:
In this paper we propose a crystalline motion of spiral-shaped polygonal curves with a tip motion as a simple model of a step motion on a crystal surface under screw dislocation. We give a tip motion and discuss the behavior of the solution curves by crystalline curvature flow with a driving force. We show that the solution curve belongs to a suitable class of spiral-shaped curves and also show a time-global existence of the spiral-shaped solutions.
2014, 7(1): 63-74 doi: 10.3934/dcdss.2014.7.63 +[Abstract](1446) +[PDF](342.6KB)
Abstract:
We shall show the existence of a solution for a nonlinear parabolic system. This system is a tumor invasion model which has the time and space dependent diffusion coefficient. In this paper, we apply an existence result for Quasi-Variational Inequalities. Quasi-Variational Inequality is a problem to find a function which satisfies a variational inequality in which the constraint depends upon the unknown function. In this paper, I shall show how to approach to our tumor invasion model by Quasi-Variational inequality, and obtain a solution for it.
2014, 7(1): 75-93 doi: 10.3934/dcdss.2014.7.75 +[Abstract](1596) +[PDF](441.8KB)
Abstract:
The main objective of this paper is to discuss solvability of the Cauchy problem of an evolution equation with subdifferentials of convex functions which is generated by unknown functions and perturbations of the form:
$u'(t) + ∂ \varphi^t(u;u(t)) + G(u(t)) \ni f(t)$   0 < t < T,      in     H.
where H is a Hilbert space, $u'=\frac{du}{dt}$, and $∂ \varphi^t(u;\cdot )$ is a subdifferential operator of convex function $\varphi^t(u;\cdot )$. The evolution equation corresponds to parabolic quasi-variational inequalities.
2014, 7(1): 95-111 doi: 10.3934/dcdss.2014.7.95 +[Abstract](1965) +[PDF](585.0KB)
Abstract:
We consider the flow and transport of chemically reactive substances (precursors) in a channel over substrates having complex geometry. In particular, these substrates are in the form of trenches forming oscillating boundaries. The precursors react at the boundaries and get deposited. The deposited layers lead to changes in the geometry and are explicitly taken into account. Consequently, the system forms a free boundary problem. Using formal asymptotic techniques, we obtain the upscaled equations for the system where these equations are defined on a domain with flat boundaries. This provides a huge gain in computational time. Numerical experiments show the effectiveness of the upscaling process.
2014, 7(1): 113-125 doi: 10.3934/dcdss.2014.7.113 +[Abstract](1631) +[PDF](341.6KB)
Abstract:
In this paper we prove the existence of a solution for a mathematical model of carbon dioxide transport in concrete carbonation process. This model is a parabolic type equation with a nonlinear perturbation such that a coefficient of the time derivative contains a non-local term depending on the unknown function itself.
2014, 7(1): 127-137 doi: 10.3934/dcdss.2014.7.127 +[Abstract](2275) +[PDF](356.6KB)
Abstract:
In this manuscript, we consider a Cahn-Hilliard/Allen-Cahn equation is introduced in [17]. We give an existence of the solution, slightly improved from [18]. We also present a stochastic version of this equation in [3].
2014, 7(1): 139-159 doi: 10.3934/dcdss.2014.7.139 +[Abstract](1777) +[PDF](286.3KB)
Abstract:
In this paper, a coupled system of two parabolic initial-boundary value problems is considered. The system presented is a one-dimensional version of the Kobayashi-Warren-Carter model of grain boundary motion [15,16], that is derived as a gradient system of a governing free energy including a weighted total variation. Due to the weighted total variation, some nonstandard terms appear in the mathematical expressions of this system, and such nonstandard terms have made the mathematical treatments to be quite delicate. Recently, a certain definition of the solution have been provided in [21], together with the solvability result. The main objective in this paper is to verify that the system reproduces the foundational rules as a gradient system of parabolic PDEs, such as smoothing effect'' and energy-dissipation''. Consequently, the existence of a special solution, called energy-dissipative solution'', will be demonstrated in the Main Theorem of this paper.
2014, 7(1): 161-176 doi: 10.3934/dcdss.2014.7.161 +[Abstract](1503) +[PDF](413.9KB)
Abstract:
We study the kinetic mean field equation on two-dimensional Brownian vortices; derivation, similarity to the DD-model, and existence and non-existence of global-in-time solution.
2014, 7(1): 177-189 doi: 10.3934/dcdss.2014.7.177 +[Abstract](1821) +[PDF](363.8KB)
Abstract:
In this paper we consider the initial boundary value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Moreover, approximate solutions for this equation may not belong to $BV$. These are difficult points for this type of equations.
We consider the type of equations under the zero-flux boundary conditions. In particular, we prove the existence and partial uniqueness of weak solutions to such problems. Our proof use the compactness theorem derived by Panov [14] and the estimate of degenerate diffusion term derived by Karlsen-Risebro-Towers [10].

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