Nonstationary ideal incompressible fluid flows: Conditions of existence and uniqueness of solutions

The problem of formulating minimal conditions on input data that can guarantee the existence and uniqueness of solutions of the boundary value problems describing non-one-dimensional ideal incompressible fluid flow is considered using as an example the initial boundary value problem in a space-time cylinder constructed on a bounded flow domain with the nonpenetration condition on its boundary (which corresponds to fluid flow in a closed vessel). The existence problems are considered only for plane flows, and the uniqueness issues for three-dimensional flows as well. The required conditions are obtained in the form of conditions specifying that the vorticity belongs to definite functional Orlicz spaces. The results are compared with well-known results. Examples are given of admissible types of singularities for which the obtained results are valid, which is a physical interpretation of these results. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 130–145, July–August, 2008.     

On the existence and uniqueness of solutions generated by equivalent linearization

The existence and uniqueness of approximate solutions generated by the generalized method of equivalent linearization is considered. For the stationary analysis of systems with harmonic or Gaussian random excitation, it is shown that even though the equivalent linear system may not be unique, a simple element-by-element substitute system exists. Furthermore, this system is at least as good as any other similarly defined substitute system.     

Global existence and uniqueness of solutions for the equations of third grade fluids

We consider the equations governing the motion of third grade fluids in . We show global existence of solutions without any smallness assumption, by assuming only that the initial velocity belongs to the Sobolev space H². The uniqueness of such solutions is also proven in dimension two.     

The existence and uniqueness of solutions of generalized variational inequalities arising from elasticity with friction

ＩｎｔｒｏｄｕｃｔｉｏｎＡｔｐｒｅｓｅｎｔ,ｔｈｅｒｅｓｅａｒｃｈｏｎｖａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｉｅｓｃａｕｓｅｄｂｙｅｌａｓｔｉｃｉｔｙｗｉｔｈｆｒｉｃｔｉｏｎａｒｅｓｔｉｌｌｒｅｓｔｒｉｃｔｅｄｔｏｔｈｅｅｑｕｉｖａｌｅｎｔｍａｘｉｍｕｍｏｒｍｉｎｉｍｕｍｅｎｅｒｇｙｐｒｉｎｃｉｐｌｅｓ .Ｓｏｍｅｆａｍｏｕｓｅｘｐｅｒｔｓ,ＧＵＯＹｏｕ＿ｚｈｏｎｇ[1],ＺＨＯＵＳｈｕ＿ｚｈｉ[2 ],Ｇ .Ｆ .ＣａｒｅｙａｎｄＪ.Ｔ .Ｏｄｅｎ[3]ａｎｄＪ.Ｔ .ＯｄｅｎａｎｄＬ .Ｃａｍｐｏｓ[4 ],ａｌｌｈａｖ…     

The existence and uniqueness of the decaying positive entire solutions for a class of semilinear elliptic equations

ＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＴＨＥＤＥＣＡＹＩＮＧＰＯＳＩＴＩＶＥＥＮＴＩＲＥＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＳＥＭＩＬＩＮＥＡＲＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳＴｉａｎＧｅｎｂａｏ（田根宝）（ＳｈａｎｇｈａｉＩｎ...     

An existence and uniqueness theorem in linear elastodynamics

Summary Laplace transform with respect to time is applied to the boundary-initial-value problem of linear elastodynamics in order to produce a simpler elliptic boundary-value problem. Existence and uniqueness of the weak solution of the latter problem are proved.

---

Sommario Si studiano le questioni di esistenza e unicità per il problema al contorno di tipo ellittico che risulta dalla trasformazione secondo Laplace rispetto al tempo del problema nei valori iniziali e al contorno della elastodinamica lineare.

     

A local existence and uniqueness theorem for a K-BKZ-fluid

We consider the three-dimensional motion of a viscoelastic liquid occupying all of space. The constitutive law is assumed to be of the form suggested by Kaye [13] and Bernstein, Kearsley & Zapas [2]. An existence and uniqueness result for solutions of the initial value problem on sufficiently short time intervals is proved using Kato's theory of quasilinear hyperbolic equations.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants Nos. MCS-8210950 and MCS-8215064.     

On uniqueness, existence and objectivity of S-R decomposition theorem

I.IntroductionThedecompositionofadeformationgradientfunctionFisafundamentalproblemforfinitedeformationtheory.Greenstraintheoryandpolardecompositiontheorywereemployedbymostscientists,However,thesetheorieshaveallsomeconsiderabledrawbacks.Inl979,ChenZhidadev…     

Stationary problem of third-grade fluids in two and three dimensions: existence and uniqueness

This paper is devoted to the stationary problem of third-grade fluids in two and three dimensions. In two dimensions, we show existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, by generalizing the method used by J.M. Bernard for the stationary problem of second-grade fluids (we deal with a polynomial of four degrees instead of two degrees). Contrary to the case of two dimensions, the resolution of the problem of third-grade fluids in three dimensions requires the physical condition |α₁+α₂|<(24νβ)^1/2. From this condition, we derive two “pseudo ellipticities” for the operator ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, where A(u) is a 3-order symmetric matrix such that tr(A(u))=0. Thus, with, in addition, a sharp estimate of the scalar product (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), we are able to prove existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, in three dimensions.

Résumé

Cet article est consacré au problème stationnaire des fluides de grade trois en dimension deux et trois. En dimension deux, nous montrons l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en généralisant la méthode utilisée par J.M. Bernard pour le problème stationnaire des fluides de grade deux (nous avons affaire à un polynôme de degré quatre au lieu de deux). Contrairement au cas de la dimension deux, la résolution du problème des fluides de grade trois en dimension trois requière la condition physique |α₁+α₂|<(24νβ)^1/2. De cette condition, nous déduisons deux “pseudo matrice” pour l’opérateur ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, où A(u) est une matice symétrique d’ordre 3 à trace nulle. De là, avec, en plus, une fine estimation du produit scalaire (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), nous sommes capables de prouver l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en dimension trois.     

On the existence and uniqueness of solutions for a class of variational inequalities with applications to the signorini problem in mechanics

In this paper,we introduce a new unified and general class of variational inequalities,and show some existence and uniqueness results of solutions for this kind of variationalinequalities.As an application,we utilize the results presented in this paper to study theSignorini problem in mechanics.     

Fundamental properties of elasto-plastic matrix and existence and uniqueness of elasto-plastic stress and displacement

Based on the assumption that the yield criterionF(,)=0 is a differentiable surface in stress space and by starting from a specially defined real Euclidean space, the fundamental properties of the elasto-plastic matrix in the incremental theory of plasticity is discussed in detail. By using these results, a convex analysis is made to prove the existence and uniqueness of 1) the distribution of incremental elasto-plastic stress for work-hardening materrials; 2) the displacement distribution for work-hardening materials. Material isotropy is assumed in all discussions of relevant problems.     

Virtual displacements principle,existence and uniqueness for elastic no tension bodies

Aim of this paper is the formulation of the virtual displacement principle, and some its applications, for bodies made of masonry like–material, i.e. the elastic no tension material (ENT). In place of the common adoption of continuously diffused smeared cracks, used to include both cracks and fracture strains, the paper considers the direct presence of discontinuities of the displacement fields, i.e. the cracks occurring in masonry bodies. The paper makes use of the notion of the new boundary of the body which includes the fractures, following the approach of Volpert and Hudjaev (Analysis in classes of discontinuous functions and equations of mathematical physics, Nijhoff, Dordrecht, 1985). In this framework the paper extends to the ENT bodies a previous formulation of the principle of virtual displacements, given by the Author, for the simpler case of the rigid no tension bodies. The consequent study of the existence and uniqueness of the admissible equilibrium concludes the paper.     

On uniqueness of solutions for secondary creep problems

Uniqueness results are established for solutions of secondary creep problems, including the effect of elastic strains, for a large class of domains subject to mixed boundary conditions. Two theorems are proved, one for quasistatic creep and one for dynamic.