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Evolutionary PDE’s in perfectly plastic fluid theory

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Evolutionary PDE’s in perfectly plastic fluid theory

Dominic Breit and Joachim Naumann

The equation of motion for an incompressible perfectly plastic fluid on a bounded domain Ω ⊂ R

d

, d = 2, 3, during the time intervall (0, T ) reads as

−∂

t

u + div σ = ∇π − f on Q := Ω × (0, T ).

Here u : Q → R

d

is the velocity field, π : Q → R the pressure, σ : Q → R

d×d

denotes the stress deviator and f : Q → R

d

an external system of volume forces. Between σ and the symmetric gradient ε(u) of the velocity field we have the following relation (constitutive law) which was introduced by von Mises in 1913 (g is the yield value)

ε(u) = 0 ⇒ |σ| ≤ g, ε(u) 6= 0 ⇒ σ = g

|ε(u)| ε(u).

We show the existence of a weak solution

(u, σ) ∈ L

1

(0, T ; BD

div

(Ω)) × L

(Q, R

d×d

)

to the equation above where the constitutive law has to be understand in a measure theoretical fashion. The space BD(Ω) denotes the class of L

1

-functions whose distributional symmetric gradient generates a bounded Radon measure.

1

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