Admissibility conditions





0 if x < ^α₂t

α if ^α₂t < x < ^1+α₂ t 1 if x ≥ ^1+α₂ t

is a weak solution of our problem, as it is a piecewise constant function and satisfies the Rankine-Hugoniot condition

λ(u^r_α− u^lα) = u^r_α 2

2

− u^r_α 2

2

, along the characteristic lines:

x(t) = α

2t and x(t) = 1 + α 2 t.

Since this is valid for every 0≤ α ≤ 1, we have infinite solutions to this problem.

In the following paragraphs we want to find additional constraints so that we can identify a unique solution among the possible infinite ones. We will see that there are a number of these conditions, many are equivalent, and we need to make a choice. We will describe the most relevant for our purposes.

2.7.2 Admissibility conditions

There are three main approaches to impose enough constraints to force the weak solution to be unique:

“Viscosity solutions”, “Entropy conditions”, and “Stability conditions”. We follow again [2] and call these conditions “admissibility conditions”.

Viscosity solutions or Vanishing viscosity. In this approach we add a “viscosity” term to our con-servation law, and we look for solutions at the limit of “zero” viscosity. Hence, given a positive , we reformulate the problem as:

u^_t+ f (u^)_x= u^_xx, (2.45)

and look for the limit as  7→ 0⁺. Thus we say that a weak solution u(x, t) is “admissible in the vanishing viscosity sense” if u^(x, t)7→ u(x, t) as  7→ 0.

This approach is usually consistent, as most conservation laws are physically obtained by assuming dissipation (a term typically proportional to the second derivative) to be negligible. However, this condition is difficult to assess in practice, and has mostly a theoretical use.

Entropy conditions. This admissibility condition takes origin from the observation that in thermo-dynamically dissipative systems the entropy is never decreasing and if the system is non-dissipative then entropy is conserved. Hence we look for a scalar function S of the system state vector u and say the entropy S(u) and the entropy flux q(u) should obey a conservation law.

We say that a scalar function S(u) : R^d 7→ R is called the “entropy” associated with the system of conservation laws (2.39) with corresponding entropy flux q(u) : R^d 7→ R if they satisfy:

J f(u) ∇ S(u) = A(u) ∇ S(u) = ∇ q(u). (2.46)

For such a function, we have:

S(u)_t+ q(u)_x =h∇ S(u), uti + h∇ q(u), uxi = h∇ S(u), (ut+Juf (u)u_x)i = 0, (2.47) showing that entropy is conserved. This is true for continuous solutions, but not so if discontinuities are present, as the following example shows.

Example 2.15. For Burgers equation with flux f (u) = u²/2, we can define an entropy function S(u) = u³ and the corresponding entropy flux q(u) = 3/4u⁴. An easy calculation shows that they satisfy the defining condition (2.46). We consider the piecewise constant weak solution for Burgers equation given by:

u(x, t) =

(u^L= 1 if x < t/2, u^R= 0 if x≥ t/2.

Given this solution, the Rankine-Hugoniot condition for equation (2.47) is not satisfied. Indeed, since the shock speed is λ = 1/2, we have:

3

4 q(u^R)− q(u^L)
6= λ S(u^R)− S(u^L)
= 1 2.

If we multiply equation (2.45) by∇ S and using the definition of entropy function, we have⁷: S(u^)_t+ q(u^)_x = ∇ S(u^)u^_xx =  [S(u^)_xx− J^x(∇ S(u^))· u^xx] .

The last term in the square brackets is always positive if S is a convex function. Multiplying by a positive and smooth test function φ with compact support and integrating by parts we have:

Z Z

[S(u^)φ_t+ q(u^)φ_x] dxdt≥ −

Z Z

S(u^)φ_xx dxdt.

7We note that

(S(u))xx= (∇ S · u^x)x=J^x(∇ S) · u^x+∇ S · u^xx.

x u

u

u

u

x u

u

u

u

FIGURE 2.17: The solution to this problem is stable if the wave that stays behind has a speed that is larger or equal than the speed of the wave ahead.

If u^ 7→ u as  7→ 0 we have:

Z Z

S(u)φ_t+ q(u)φ_x dxdt≥ 0

for φ > 0. Again using integration by parts we can show that the previous statement implies S(u)t+ q(u)_x ≤ 0.

Hence we say that a weak solution u(x, t) of (2.39) is “entropy admissible” if:

S(u)_t+ q(u)_x ≤ 0.

Using the divergence theorem, it can be shown that the previous inequality holds if and only if:

λS(u^R)− S(u^L) ≥ q(u^R)− q(u^L).

Stability (Liu) conditions. We start from a scalar nonlinear conservation law. Consider a piecewise constant weak solution jumping from u^Lto u^Racross the characteristics (as in (2.41)). We perturb our problem by adding an intermediate state u^L≤ u^∗ ≤ u^R(or vice-versa). Then the shock is subdivided into two parts moving with speed given by the Rankine-Hugoniot conditions (see Figure 2.17). The distance between the waves of the original and the perturbed problems does not increase in time only if the speed of the shock behind is larger or equal to the speed of the shock ahead. This is obtained if and only if:

f (u^∗)− f(u^L)

u^∗− u^L ≥ f (u^R)− f(u^∗)

u^R− u^∗ for all u^∗ ∈ [u^L, u^R].

The previous equation identifies secant lines in the graph of f (u). In practice, if u^L < u^Rthe graph of f should remain above the secant line (Figure 2.18, left), while if u^L > u^Rthe graph of f should stay below. This must hold also for any internal state u^∗ ∈ [u^L, u^R].

u f

u^l u^∗ u^r u

f

u^l u^∗ u^r

FIGURE 2.18: Geometrical interpretation of the stability based admissibility condition for a scalar conservation law.

x t

x t

FIGURE2.19: Geometrical interpretation of the Lax admissibility condition for a scalar conservation law: Admissible shock (left) and non-admissible shock (right).

In the case of systems, the situation is obviously more complicated. In practice stability implies that the same statement holds for any intermediate condition u^∗. In other words, a shock with left and right states given by u^Land u^Rsatisfies the admissibility condition if its speed is less than or equal to the speed of every smaller shock connecting u^Lwith u^∗. This condition due to Liu [10] has been shown to identify solutions equivalent to viscosity solutions and is called the Liu admissibility condition.

Lax admissibility conditions. A final admissibility criterion due to Lax [8] must be mentioned because of its wide spread use in the numerical solution of systems of conservation laws. A shock of the i-th family is said to satisfy the “Lax admissibility criterion” if it moves with a speed λ_i(u^L, u^R) such that:

λ_i(u^L)≥ λi(u^L, u^R)≥ λi(u^R).

Consider again a piecewise smooth (constant) weak solution u(x, t) of our system with a discontinuity along the curve x = γ(t), across which u assumes the left and right states u^L and u^R. The i-th wave must travel with a speed dx/dt = ˙γ(t) = λi(u^L, u^R) given by the i-th eigenvalue of the average matrix A(u^L, u^R) defined in (2.43). This requires that the characteristic lines run into the shock line from both sides. The geometrical interpretation is reported in Figure 2.19 showing on the left panel an admissible shock and on the right panel a non-admissible one.