Order Form - FENOMEC

22 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
Finally, if a nonlinear viscosity µ(w) is also taken into account, we arrive at a general
model including viscosity and capillarity effects:
∂ t v − ∂ x Σ(w, w x ,w xx )= ( µ(w) v x
)x ,
∂ t w − ∂ x v =0.
(4.18)
The total energy
E(w, v, w x ):=e(w, w x )+v 2 /2
again plays the role of a mathematical entropy. We find
(
∂ t E(w, v, w x ) − ∂ x Σ(w, wx ,w xx ) v ) (
∂e
)
= v x (w, w x ) + ( µ(w) vv x
∂w x x
)x − µ(w) v2 x.
Once more, the entropy inequality similar to (4.14) could be obtained.
Finally, let us discuss the properties of the internal energy function e. A standard
choice in the literature is for e to be quadratic in w x . (Linear term should not appear
because of the natural invariance of the energy via the transformation x ↦→ −x.)
Setting, for some positive capillarity coefficient λ(w),
e(w, w x )=ε(w)+λ(w) w2 x
2 , (4.19)
the total stress decomposes as follows:
Σ(w, w x ,w xx )=σ(w)+λ ′ (w) w2 x
2 − (λ(w) w x) x , σ(w) =ε ′ (w). (4.20)
The equations in (4.18) take the form
(
∂ t v − ∂ x σ(w) = λ ′ (w) w2 x
2 − ( )
λ(w) w x +
)x
µ(w) v x
x
)x , (4.21)
∂ t w − ∂ x v =0.
In this case we have
)
(ε(w)+ v2
2 + λ(w) w2 x
2
= ( µ(w) vv x
)x − µ(w) v2 x +
t − ( σ(w) v ) x
(
v λ′ (w)
wx 2 − v ( )
λ(w) w x
2
+ v x x λ(w) w x .
)x
Under the simplifying assumption that the viscosity and capillarity are both constants,
we can recover Example 4.6 above.
□
5. Kinetic relations and traveling waves
We return to the general discussion initiated in Section 3 and we outline an important
standpoint adopted in this course for the study of (1.1). The weak solutions of
interest are primary those generated by an augmented model of the general form
(3.1). When small physical parameters accounting for the viscosity, heat conduction,
or capillarity of the material are negligible with respect to the scale of hyperbolic
features, it is desirable to replace (3.1) with the hyperbolic system of conservation
laws (1.1). Since the solutions of the Cauchy problem associated with (1.1) are not
unique, one must determine suitable admissibility conditions which would pick up
the solutions of (1.1) realizable as limits of solutions of (3.1) by incorporating some