On evolution inequalities of Bingham type in three dimensions, II

Naumann, Joachim; Wulst, Michael

doi:10.1016/0022-247x(79)90046-5

Cited by 4 publications

(6 citation statements)

References 8 publications

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“…The assertion for the regularized problem is PROPOSITION 3.3. For each e > 0, there is a scalar function p£(x,t) and a unique function ue(x,t) such that 3 dtu£ = pAu£ + peAAu£ + g^ dj{(e + Du{u£))~1/2Dij(u£)} Again by (3)(4)(5)(6)(7)(8)(9) and the fact that uoE(x) EVi, we have + ^2uEjdjUe -f in fi x (0, T).…”

Section: Uo(x)eg /Ec([0t];[l2(fi)]3) and Dtf E L2(0t;[w~^2(n)]3)mentioning

confidence: 99%

“…In this paper we prove the existence of local solutions of (0-1) to (0-4) which are similarly regular under the same assumptions on the data as in [9]. Since (0-1) reduces to the Navier-Stokes equations when g = 0, we are tempted to utilize the known techniques of analysis for the Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

“…They [2,3] also obtained more regular solutions in a two dimensional domain. For a variant of Bingham fluid, Naumann and Wulst [9] established the existence of the same kind of regular solutions in a three dimensional domain through a different method. They [9] assumed that the initial data belong to a special class of stationary states and essentially used the assumption of "averaged nonlinear viscosity."…”

Section: Introductionmentioning

confidence: 99%

“…For a variant of Bingham fluid, Naumann and Wulst [9] established the existence of the same kind of regular solutions in a three dimensional domain through a different method. They [9] assumed that the initial data belong to a special class of stationary states and essentially used the assumption of "averaged nonlinear viscosity." Our model described by (0-1) does not satisfy this assumption and the result of [9] cannot be applied.…”

Section: Introductionmentioning

confidence: 99%

“…They [9] assumed that the initial data belong to a special class of stationary states and essentially used the assumption of "averaged nonlinear viscosity." Our model described by (0-1) does not satisfy this assumption and the result of [9] cannot be applied.…”

Section: Introductionmentioning

confidence: 99%

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On the initial-boundary value problem for a Bingham fluid in a three-dimensional domain

Kim¹

1987

Trans. Amer. Math. Soc.

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ABSTRACT. The initial-boundary value problem associated with the motion of a Bingham fluid is considered.The existence and uniqueness of strong solution is proved under a certain assumption on the data. It is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time-periodic 0. Introduction.The purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a Bingham fluid in a bounded three dimensional domain. A Bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached. Since the motion is governed by two entirely different stress-strain relations depending on the state of stresses, the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the rigid zone.The initial-boundary value problem we shall study is formulated asfor each test function w such that V ■ w = 0 in Yl and w = 0 on dYl,Here, fi is a bounded domain in R3 with smooth boundary dYl, u(x, t) denotes the velocity of the fluid and f(x, t) stands for external force. We assume that the density, the yield limit and the viscosity are positive constants. In particular, the

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