Evolutionary Inequalities Research Articles

Sur les solutions bornées et presque-periodiques des équations et inéquations d'evolution

We show some existence et uniqueness theorems for the bounded and almost periodic solution of some non linear equations and inequalities of evolution.

Degenerate variational inequalities of evolution

On etudie des inegalites variationnelles de la forme Re ≥0 pour tout v∈K, By(0)=Bu 0 , ou B est lineaire mais peut s'annuler, A est non lineaire et K est un ensemble convexe ferme

A parabolic-elliptic variational inequality

An existence and uniqueness result is proved for a variational inequality of evolution. The problem consists of a nonlinear parabolic/elliptic differential equation for which Dirichlet, Neuman and Signorini boundary conditions are posed. The existence is obtained using a regularization process, while uniqueness is based on L1-contractiveness of the solution semigroup.

On evolution inequalities of Navier-Stokes type in three dimensions

Nel presente lavoro si dimostra un teorema sull'esistenza ed un teorema di regolarita per una soluzione del problema $$\begin{gathered} u' + Au + B(u, u) + \partial \varphi (u) \mathrel\backepsilon f, \hfill u(0) = u_0 . \hfill \end{gathered} $$

Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities

AbstractThe change of variable for the temperature Θ in the one‐phase Stefan problem leads to the evolution inequality, (ut – Δu – f)(v – u) ⩾ 0 for all regular v ⩾ 0, where u ⩾ 0 is required. This inequality is to hold over a space‐time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ∂ Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {um}. The sequence {um} may be interpolated linearly in t to obtain an approximation UΔt of u. The following results are obtained in this paper: (i) a two‐sided weak maximum principle for um – um‐1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of UΔt and ∇UΔt, to u and ∇u, respectively, on D; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence xΔt, where xΔt interpolates xΔt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.

On a Class of First Order Evolution Inequalities Arising in Heat Conduction with Memory

This paper deals with the evolution problem \[u' + a_0 u + \int_0^t {a(t - s)u(s)ds} + \int_0^t {b(t - s)Bu} (s)ds + \partial \varphi (u) \ni f,\quad u(0) = u_0 ,\] on an arbitrary interval $[0,T]$, where $a_0 \in \mathbb{R}$ and $a,b:[0,T] \to \mathbb{R}$ are given functions; B is a linear bounded mapping from a Hilbert space V into its dual, while $\partial \varphi $ denotes the subdifferential mapping of a proper, convex and lower semicontinuous functional $\varphi :V \to ( - \infty , + \infty ]$, We prove an existence and uniqueness theorem for a solution u to the above problem. The existence is established by combining the Galerkin method with a regularization of the functional $\varphi $. An application of the abstract result to a Volterra integrodifferential equation arising from the theory of heat conduction in materials with memory is also given.

Sur une inequation d’evolution parabolique du type II, associee a un probleme d’asserviment thermique

We consider an evolution unilateral problem with time-dependent obstacles. The problem studies the temperature-field in an open bounded set filled with a continuous medium. We obtain a convenient regularity theorem and using the finite affine triangular element method, we discretize the corresponding evolution inequality. After all we obtain, via the regularity theorem, an estimate of the discretization error.

On evolution inequalities of Bingham type in three dimensions, II

On evolution inequalities of Bingham type in three dimensions, II

On evolution inequalities of Bingham type in three dimensions, I

On evolution inequalities of Bingham type in three dimensions, I

On evolution inequalities of a modified Navier-Stokes type. II

The present part of the paper continues the study of the abstract evolution inequality from the first part. Theorem 1 states the existence and uniqueness of a weak solution to the evolution inequality under consideration. The proof is based on the method of approximation of the weak solution by a sequence of strong solutions. Theorem 2 yields two regularity results for the strong solution.

Periodic solutions to evolution inequalities related to the Bingham problem in two dimensions

Periodic solutions to evolution inequalities related to the Bingham problem in two dimensions

Periodic solutions to certain evolution inequalities

Periodic solutions to certain evolution inequalities

A Convexity Result for Weak Differential Inequalities

In this note we present a natural “weak” form of a certain convexity estimate for evolution inequalities as given in Agmon-Nirenberg’s paper [1], p. 139 (see also A. Friedman [2], Theorem 4.2 and 4.3). Our proof will follow that given in [1] and [2] with the natural modifications due to the enlargement of the class of solutions which are taken into account.

On the support of the solution of some variational inequalities of evolution

On the support of the solution of some variational inequalities of evolution

Transfert de chaleur dans un fluide de Bingham dont la viscosité dépend de la température

The paper studies the flow of a so-called Bingham fluid, taking into account the variation of viscosity with temperature. The problem is modelled using the technique of variational inequalities of evolution. One proves the existence of a solution in case of a 2-dimensional flow. Questions of uniqueness, and of existence and uniqueness for 3-dimensional flows, are open. The proof relies on fixed-point theory, previous results of the authors for Bingham flows when viscosity does not depend on the temperature and a regularity theorem of Grisvard for linear parabolic equations.

PARTIAL DIFFERENTIAL INEQUALITIES

Contents Introduction Chapter 1. Variational elliptic inequalities § 1.1. First example § 1.2. Variational inequalities § 1.3. Variational inequalities and many-valued operators § 1.4. Examples § 1.5. The method of penalization § 1.6. The method of regularization § 1.7. Interpretation of variational inequalities and multipliers § 1.8. Regularity § 1.9. Comparison and convexity properties of solutions Chapter 2. Parabolic evolution inequalities (I) § 2.1. First examples § 2.2. The variational formulation. The strong solution § 2.3. The variational formulation. The weak solution § 2.4. Examples § 2.5. The method of elliptic regularization § 2.6. The method of penalization § 2.7. Some generalizations § 2.8. Quasi-stationary problems § 2.9. Regularity § 2.10. Remarks Chapter 3. Parabolic evolution inequalities (II). The flow of a non-Newtonian fluid § 3.1. Statement of the problem § 3.2. The main results § 3.3. The principle of the proof Chapter 4. Inequalities of the second order with respect to t § 4.1. First example § 4.2. The general variational problem § 4.3. Examples § 4.4. Methods for existence proofs Chapter 5. Variational inequalities for Maxwell operators § 5.1. Statement of the problem § 5.2. Results Chapter 6. Other problems § 6.1. Coupled phenomena § 6.2. A problem with a decay § 6.3. Behaviour at infinity. Almost periodic solution Chapter 7. General remarks on the numerical approximation of solutions of inequalities § 7.1. Discrete schemes for evolution inequalities § 7.2. Analysis of stability § 7.3. The solution of stationary problems Chapter 8. Singular layer phenomena § 8.1. Generalities § 8.2. An example Chapter 9. Open problems References

Lower bounds and uniqueness for solutions of evolution inequalities in a Hilbert space

Publisher Summary This chapter discusses lower bounds and uniqueness for solutions of evolution inequalities in a Hilbert space. The form of the inequality will offer a wider range of applicability of results to partial differential equations. Let H be a Hubert space with inner product (,) and norm ∥·∥. By C[J,H], one denotes the space of continuous functions from the real interval J into the Hubert space H. If u ∈ C [J, H], then denotes the derivative of u with respect to t in the strong topology of H. This obtains general lower bounds and two uniqueness theorems for the solutions on an interval J of the real line R. One shall use the inequality m(t)≥∥u(t)∥2 , which is valid as P(u(t)) ≥0.

Remarks on evolution inequalities

Remarks on evolution inequalities