A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The use of the thermoelastic martensitic transformation and its reverse transformation has recently been proposed and demonstrated for several active control ap plications. However, the present constitutive models have lacked several important funda mental concepts that are essential for many of the proposed intelligent material system ap plications such as shape memory hybrid composites.A complete, unified, one-dimensional constitutive model of shape memory materials is developed and presented in this paper. The thermomechanical model formulation herein will investigate important material characteristics involved with the internal phase transformation of shape memory alloys. These characteristics include energy dissipation in the material that governs the damping behavior, stress-strain-temperature relations for pseudoelasticity, and the shape memory effect. Some numerical examples using the model are also presented.

One-Dimensional Thermomechanical Constitutive Relations for Shape Memory Materials

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

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Nonlinear partial differential equations with applications

Variational models for microstructure and phase transitions

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

1. Some Mathematical Tools.- 1.1 Measure and Integration.- 1.2 Function Spaces.- 1.3 Nonlinear Equations.- 1.4 Ordinary Differential Equations.- 2. Hysteresis Operators.- 2.1 Basic Examples.- 2.2 General Hysteresis Operators.- 2.3 The Play Operator.- 2.4 Hysteresis Operators of Preisach Type.- 2.5 Hysteresis Potentials and Energy Dissipation.- 2.6 Hysteresis Counting and Damage.- 2.7 Characterization of Preisach Type Operators.- 2.8 Hysteresis Loops in the Prandtl Model.- 2.9 Hysteresis Loops in the Preisach Model.- 2.10 Composition of Preisach Type Operators.- 2.11 Inverse and Implicit Hysteresis Operators.- 2.12 Hysteresis Count and Damage, Part II.- 3. Hysteresis and Differential Equations.- 3.1 Hysteresis in Ordinary Differential Equations.- 3.2 Auxiliary Imbedding Results.- 3.3 The Heat Equation with Hysteresis.- 3.4 A Convexity Inequality.- 3.5 The Wave Equation with Hysteresis.- 4. Phase Transitions and Hysteresis.- 4.1 Thermodynamic Notions and Relations.- 4.2 Phase Transitions and Order Parameters.- 4.3 Landau and Devonshire Free Energies.- 4.4 Ginzburg Theory and Phase Field Models.- 5. Hysteresis Effects in Shape Memory Alloys.- 5.1 Phenomenology and Falk's Model.- 5.2 Well-Posedness for Falk's Model.- 5.3 Numerical Approximation.- 5.4 Complementary Remarks.- 6. Phase Field Models With Non-Conserving Kinetics.- 6.1 Auxiliary Results from Linear Elliptic and Parabolic Theory.- 6.2 Well-Posedness of the Caginalp Model.- 6.3 Well-Posedness of the Penrose-Fife Model.- 6.4 Complementary Remarks.- 7. Phase Field Models With Conserved Order Parameters.- 7.1 Well-Posedness of the Caginalp Model.- 7.2 Well-Posedness of the Penrose-Fife Model.- 8. Phase Transitions in Eutectoid Carbon Steels.- 8.1 Phenomenology of the Phase Transitions.- 8.2 The Mathematical Model.- 8.3 Well-Posedness of the Model.- 8.4 The Jominy Test: A Numerical Study.

Hysteresis and Phase Transitions

This monograph provides a comprehensive and accessible introduction to the optimization of elliptic systems. This area of mathematical research, which has many important application in science and technology, has experienced an impressive development during the last two decades. This monograph aims to address some of the pressing unsolved questions in the field. The exposition concentrates along two main directions: the optimal control of linear and nonlinear elliptic equations, and problems involving unknown and/or variable domains. Throughout this monograph, the authors elucidate connections between seemingly different types of problems. One basic feature is to relax the needed regularity assumptions as much as possible in order to include larger classes of possible applications. The book is organized into six chapters that give a gradual and accessible presentation of the material, and a special effort is made to present numerous examples.

This monograph is addressed to a large readership, primarily to graduate students and researchers working in this field of mathematics. Much of this material will prove useful also for scientists from other fields where the optimization of elliptic systems occurs, such as physics, mechanics, and engineering.

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Optimization of Elliptic Systems

https://www-dimat.unipv.it/pier/papers/Colli-Gilardi-Sprekels.pdf

On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins–Daruud et al in Hawkins–Daruud et al (2011 Int. J. Numer. Math. Biomed. Eng. 28 3–24). The model consists of a Cahn–Hilliard equation for the tumor cell fraction coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side of the equation for σ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Frechet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

https://iopscience.iop.org/article/10.1088/1361-6544/aa6e5f/pdf

Optimal distributed control of a diffuse interface model of tumor growth

A boundary control problem for the viscous Cahn---Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

Jürgen Sprekels

Papers

Hysteresis and Phase Transitions

Optimization of Elliptic Systems

On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential

Optimal distributed control of a diffuse interface model of tumor growth

A Boundary Control Problem for the Viscous Cahn---Hilliard Equation with Dynamic Boundary Conditions