Jindřich Nečas

Solution of variational inequalities in mechanics

Etude des problemes unilateraux pour les fonctions scalaires. Contact unilateral de corps elastiques. Problemes de la theorie de la plasticite

Direct methods in the theory of elliptic equations

1.Introduction to the problem.- 2.Sobolev spaces.- 3.Exitence, Uniqueness of basic problems.- 4.Regularity of solution.- 5.Applications of Rellichs inequalities and generalization to boundary value problems.- 6.Sobolev spaces with weights and applications to the boundary value problems.- 7.Regularity of solutions in case of irregular domains and elliptic problems with variable coefficients.

Injectivity and self-contact in nonlinear elasticity

Let Ω be a bounded open connected subset of ℝ3 with a sufficiently smooth boundary. The additional condition ∫ det ▽ψ dx ≦ vol ψ(Ω) is imposed on the admissible deformations ψ: ¯Ω → ℝ of a hyperelastic body whose reference configuration is ¯Ω. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Balls theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W1,p(Ω;ℝ3), p > 3, that is injective almost everywhere.

Young Measure‐Valued Solutions for Non-Newtonian Incompressible Fluids1

For the model of a nonlinear bipolar fluid, in which the highest order viscosity vanishes, and the viscous part of the stress tensor satisfies a growth condition of the form the rate of strain tensor, we demonstrate the existence of Young-measure valued solutions for these solutions are proven to be weak solutions for and for and unique regular weak solutions for

ON THE NON-NEWTONIAN INCOMPRESSIBLE FLUIDS

The Navier-Stokes equations can be included as a special case into the class of non-Newtonian incompressible fluids with the nonlinear stress tensor τ=τ(e), the components of which satisfy the p-growth condition. Measure-valued solutions already exist for p>2n/(n+2). For the space periodic problem, the existence of the weak solution is then obtained for p>3n/(n+2). These solutions are regular and unique for p≥1+2n/(n+2).

Global solution to the compressible isothermal multipolar fluid

Abstract The global existence of weak solutions of the initial boundary value problem in bounded domains to the system of partial differential equations for viscous compressible isothermal bipolar and multipolar fluids is proved. Some other properties as cavitation, regularity up to the strong solution and uniqueness are discussed.

A Dynamic Problem of Thermoelasticity

A space periodic problem of nonlinear thermoelasticity is considered. For an elastic, linear, isotropic, homogeneous, nonviscous body in small geometry, we obtain a nonlinear system of equations. For small coefficient of the heat extension a we find a time global weak solution of the initial-value problem. The smallness of a is independent of the length of the time interval and of the datas. The space periodicity of the solution is related to the absence of reflected waves. A mixed problem for a bounded domain, even with a smooth boundary, seems to be an open problem. Our work is closely related to that by J. Necas L5J and by J. Necas, A. Novotn and V. Sverak L6J.

Some qualitative properties of the viscous compressible heat conductive multipolar fluid

This paper has been devoted to the solvability of the initial boundary value problem in bounded domains in two or three dimensions to the evolution equations of viscous multipolar compressible heat conductive fluid. Physical background to this problem is discussed in ref. 6. They were able to prove solvability for the polarity k {>=} 4. They will generalize these results to the case k {>=} 3. They will use the same method. The crucial point is a more detailed study of the characteristics to the continuity equation, which makes it possible to improve estimates of the density and consequently to prove existence of strong solutions. Further they deal with the uniqueness of these solutions. In the last section they derive some important estimates, energy and entropy estimates included, which has to be fulfilled by any solution.

Convergence of Finite Elements for Transonic Potential Flows

Using new compactness results for generalized positive functionals on the Sobolev space

Problems of the Theory of Plasticity

W^{1,p}