Gregory Seregin

Variational methods for problems from plasticity theory and for generalized Newtonian fluids

Introduction 1 Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity 1.0. Preliminaries 1.1. The classical boundary value problem for the equilibrium state of a perfect elastoplastic body and its primary functional formulation 1.2. Relaxation of convex variational problems in non reflexive spaces. General construction 1.3. Weak solutions to variational problems of perfect elastoplasticity 2 Differentiability properties of weak solutions to boundary value problems in the deformation theory of plasticity 2.0. Preliminaries 2.1. Formulation of the main results 2.2. Approximation and proof of Lemma 2.1.1 2.3. Proof of Theorem 2.1.1 and local estimate of Caccioppoli-type for the stress tensor 2.4. Estimates for solutions of certain systems of PDEs with constant coeffcients 2.5. The main lemma and its iteration 2.6. Proof of Theorem 2.1.2 2.7. Open Problems 2.8. Remarks on the regularity of minimizers of variational functionals from the deformation theory of plasticity with power hardening Appendix A A.1 Density of smooth functions in spaces of tensor-valued functions A.2 Density of smooth functions in spaces of vector-valued functions A.3 Some properties of the space BD A.4 Jensens inequality 3 Quasi-static fluids of generalized Newtonian type 3.0. Preliminaries 3.1. Partial C1 regularity in the variational setting 3.2. Local boundedness of the strain velocity 3.3. The two-dimensional case 3.4. The Bingham variational inequality in dimensions two and three 3.5. Some open problems and comments concerning extensions 4 Fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening law 4.0. Preliminaries 4.1. Some functions spaces related to the Prandtl-Eyring fluid model 4.2. Existence of higher order weak derivatives and a Caccioppoli-type inequality 4.3. Blow-up: the proof of Theorem 4.1.1 for n=3 4.4. The two-dimensional case 4.5. Partial regularity for plastic materials with logarithmic hardening 4.6. A general class of constitutive relations Appendix B B.1 Density results Notation and tools from functional analysis

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

Abstract. We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.

A Certain Necessary Condition of Potential Blow up for Navier-Stokes Equations

We show that a necessary condition for T to be a potential blow up time is lim t↑T ‖v(·, t)‖L3 = ∞. 1991 Mathematical subject classification (Amer. Math. Soc.): 35K, 76D.

Regularity results for the quasi–static Bingham variational inequality in dimensions two and three

Abstract. We consider the variational inequality describing the stationary flow of a Bingham type fluid in bounded domains. Differentiability properties of weak solutions in suitable energy spaces providing existence theorems are studied. We suppose that the volume forces belong to classes of Morrey type and generalize our previous regularity results concerning slow, steady–state flow of Bingham fluids.

Variational methods for fluids of Prandtl–Eyring type and plastic materials with logarithmic hardening

We study the slow steady-state flow of a fluid of Prandtl-Eyring type and prove (partial) regularity of the strain velocity by investigating an appropriate variational problem. We further discuss local minimizers of variational integrals which occur in the theory of plasticity with logarithmic hardening. For this model we show that the deformation gradient in the three-dimensional case is smooth up to a closed set of vanishing Lebesgue measure. The paper also presents an introduction into various function spaces which are needed to formulate the problems. Copyright

Some Remarks on Non-Newtonian Fluids Including Nonconvex Perturbations of the Bingham and Powell–Eyring Model for Viscoplastic Fluids

We consider quasi-static flows of certain viscoplastic materials for which the velocity field v can be found as a minimizer of the functional in classes of functions u : ℝn ⊃ Ω → ℝn satisfying div u = 0 and also the appropriate boundary conditions. The density ω is characteristic for the material under consideration and ℰv denotes the symmetric gradient of v. In case of a Bingham fluid we have for example ω(ℰv) = η|ℰv|2 + g|ℰv| with positive constants η and g. We also consider various perturbations of ω which are not assumed to be convex so that we have to study the relaxed variational problem. Our main result states that in all cases the symmetric derivative of the velocity field is a locally bounded function.

Some Estimates near the Boundary for Solutions to the Nonstationary Linearized Navier–Stokes Equations

In the present paper, new local estimates near the boundary are established for solutions to the nonstationary linearized Navier–Stokes equations are established. Bibliography: 8 titles.

Lecture notes on regularity theory for the Navier-Stokes equations

Preliminaries Linear Stationary Problem Non-Linear Stationary Problem Linear Non-Stationary Problem Non-Linear Non-Stationary Problem Local Regularity Theory for Non-Stationary Navier-Stokes Equations Behaviour of L3-Norm Appendix A: Backward Uniqueness and Unique Continuation Appendix B: Lemarie-Riesset Local Energy Solutions

A necessary condition of potential blowup for the Navier–Stokes system in half-space

Assuming that T is a potential blow up time for the Navier–Stokes system in