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**Mathematical problems in viscoelasticity.**
*(English)*
Zbl 0719.73013

Pitman Monographs and Surveys in Pure and Applied Mathematics, 35. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 273 p. £35.00 (1987).

To quote from the introduction, “the subject of this book is the mathematical analysis of equations which model the motions of materials with memory”. Such materials are polymers, polymer solutions and suspensions and their behavior is in general described by systems of nonlinear partial integro-differential equations of second or third order. A scalar one-dimensional linear version of such equations is
\[
u_{t}(x,t)=\eta u_{xxt}(x,t)+\beta u_{xx}(x,t)+\int^{t}_{- \infty}m(t-s)(u_{xx}(x,t)-u_{xx}(x,s))ds,
\]
where \(\eta\),\(\beta\geq 0\), \(m(\cdot)>0\), \(m'(\cdot)\leq 0\). Materials described by generalizations of this model equation can be fluids \((\beta =0)\) or solids \((\beta >0)\), and the equation can be of parabolic \((\eta >0)\), hyperbolic \((\eta =0\), \(m(0)<\infty)\) or intermediate character \((\eta =0\), \(m(0)=\infty).\)

The book is mainly concerned with questions of existence in various solution classes and of properties of solutions for these equations. Other topics, such as numerical simulation or parameter identification, are mentioned, but do not receive detailed treatment. Some equations of central importance are discussed several times, illustrating different methods for their solution and allowing a comparison of these methods.

The book is organized in six chapters. In the first chapter, the mathematical framework for the description of deformations of three- dimensional bodies is given, classes of constitutive equations for viscoelastic materials are presented, and special classes of motions are discussed for which the equations of motion reduce to scalar equations for functions of one space variable. Chapter 2 deals with properties and the development of singularities. In Chapter 3, local (in time) existence results are derived, using suitable higher-order energy estimates that are derived from differentiated versions of the equations and contraction arguments. This technique is first illustrated by problems for elastic materials and then applied for initial-boundary value problems that arise in viscoelasticity of integral type and in incompressible elasticity. In Chapter 4, techniques for the construction of global solutions are presented, leading to existence results for small smooth data (i.e. to stability results for rest states) and to large data results in a special case. In Chapter 5, methods of semigroup theory are reviewed and applied to questions of local existence and uniqueness, again for a variety of problems. The short last chapter deals with the existence of steady flows of certain non-Newtonian fluids under smallness assumptions on the data and with inflow boundary conditions.

The bibliography contains 208 references, mostly to research publications of the past two decades or so, but also to background material on mathematical analysis and partial differential equations and to papers of historical significance.

The book is mainly concerned with questions of existence in various solution classes and of properties of solutions for these equations. Other topics, such as numerical simulation or parameter identification, are mentioned, but do not receive detailed treatment. Some equations of central importance are discussed several times, illustrating different methods for their solution and allowing a comparison of these methods.

The book is organized in six chapters. In the first chapter, the mathematical framework for the description of deformations of three- dimensional bodies is given, classes of constitutive equations for viscoelastic materials are presented, and special classes of motions are discussed for which the equations of motion reduce to scalar equations for functions of one space variable. Chapter 2 deals with properties and the development of singularities. In Chapter 3, local (in time) existence results are derived, using suitable higher-order energy estimates that are derived from differentiated versions of the equations and contraction arguments. This technique is first illustrated by problems for elastic materials and then applied for initial-boundary value problems that arise in viscoelasticity of integral type and in incompressible elasticity. In Chapter 4, techniques for the construction of global solutions are presented, leading to existence results for small smooth data (i.e. to stability results for rest states) and to large data results in a special case. In Chapter 5, methods of semigroup theory are reviewed and applied to questions of local existence and uniqueness, again for a variety of problems. The short last chapter deals with the existence of steady flows of certain non-Newtonian fluids under smallness assumptions on the data and with inflow boundary conditions.

The bibliography contains 208 references, mostly to research publications of the past two decades or so, but also to background material on mathematical analysis and partial differential equations and to papers of historical significance.

### MSC:

74Hxx | Dynamical problems in solid mechanics |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

76A10 | Viscoelastic fluids |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Q72 | Other PDE from mechanics (MSC2000) |

45K05 | Integro-partial differential equations |

35L70 | Second-order nonlinear hyperbolic equations |

35K55 | Nonlinear parabolic equations |

74D05 | Linear constitutive equations for materials with memory |

74D10 | Nonlinear constitutive equations for materials with memory |