35.10. Convexity in hyperbolic problems

Convexity of the entropy function and the flux are assumption used discussing physical solutions in hyperbolic problems.

Convexity of the entropy function, \(\eta''(u) \ge 0\), and the flux \(F\), F’’(u)$$, is a requirement for the entropy inequality.

Next section discusses problems with non-convex fluxes, and the Olenik-criterion.

35.10.1. Examples

Burgers’ equation

• Flux: \(F(u) = \frac{u^2}{2}\)

• Entropy: \(\eta(u) = \frac{u^2}{2}\)

• Entropy flux: \(q(u) = \frac{u^3}{3}\)

so that the compatibility condition \(q'(u) = F'(u) \eta'(u)\) is satisfied.

The flux is convex, \(F''(u) = 1 > 0\); the entropy function is convex as well, \(\eta''(u) = 1 > 0\).

P-system

\[\begin{split} \partial_t \begin{bmatrix} \rho \\ m \end{bmatrix} + \partial_x \begin{bmatrix} m \\ \frac{m^2}{\rho} + \rho a^2 \end{bmatrix} = \partial_x \begin{bmatrix} 0 \\ \mu \partial_x \left( \frac{m}{\rho} \right) \end{bmatrix} \end{split}\]

• Flux: \(F(\mathbf{u}) = \begin{bmatrix} m \\ \frac{m^2}{\rho} + \rho a^2 \end{bmatrix} \)

• Entropy: \(\eta(\mathbf{u}) = \frac{m^2}{2 \rho} + a^2 \rho \ln \frac{\rho}{\rho_0}\)

• Diffusion flux: \(\begin{bmatrix} 0 & 0 \\ - \mu \frac{m}{\rho^2} & \mu \frac{1}{\rho} \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ m \end{bmatrix}\)

Flux. First component

\[\begin{split}\nabla_{\mathbf{u}} F_1 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\end{split}\]

\[\nabla_{\mathbf{u} \mathbf{u}} F_1 = \mathbf{0}_2 \ .\]

Second component

\[\begin{split}\nabla_{\mathbf{u}} F_2 = \begin{bmatrix} a^2 - \frac{m^2}{\rho^2} \\ 2 \frac{m}{\rho} \end{bmatrix}\end{split}\]

\[\begin{split}\nabla_{\mathbf{u} \mathbf{u}} F_2 = \begin{bmatrix} 2 \frac{m^2}{\rho^3} & - \frac{2 m}{\rho^2} \\ - \frac{2 m}{\rho^2} & \frac{2}{\rho} \end{bmatrix} \ .\end{split}\]

Entropy. Gradient

\[\begin{split}\nabla_{\mathbf{u}} \eta = \begin{bmatrix} - \frac{m^2}{2 \rho^2} + a^2\left( 1 + \ln \frac{\rho}{\rho_0} \right) \\ \frac{m}{\rho} \end{bmatrix}\end{split}\]

and Hessian

\[\begin{split}\nabla_{\mathbf{u} \mathbf{u}} \eta = \begin{bmatrix} \frac{m^2}{\rho^3} + \frac{a^2}{\rho} & - \frac{m}{\rho^2} \\ - \frac{m}{\rho^2} & \frac{1}{\rho} \end{bmatrix}\end{split}\]

Compatibility of diffusion and entropy.

\[\begin{split} \nabla_{\mathbf{u} \mathbf{u}} \eta \cdot \mathbf{D} = \mu \begin{bmatrix} \frac{m^2}{\rho^4} & -\frac{m}{\rho^3} \\ - \frac{m}{\rho^3} & \frac{1}{\rho^2} \end{bmatrix} \end{split}\]

This matrix is semi-definite positive with one zero eigenvalue (associated with exact conservation and no dissipation in mass equation), and the other eigenvalue

Euler equations for compressible flows

Navier-Stokes equations

\[\begin{split}\begin{aligned} & \partial_t \rho + \nabla \cdot ( \rho \mathbf{u} ) = 0 \\ & \rho D_t \mathbf{u} = \nabla \cdot \mathbb{T} \\ & \rho D_t e^t = \nabla \cdot ( \mathbb{T} \cdot \mathbf{u} ) - \nabla \cdot \mathbf{q} \ , \end{aligned}\end{split}\]

with \(\mathbb{T} = -p \mathbb{I} + 2 \mu \mathbb{D} + \lambda ( \nabla \cdot \mathbf{u} ) \mathbb{I}\) the constitutive equation for Newtonian fluids, and \(\mathbf{q} = - k \nabla T\) if Fourier law holds.

The conservative form of the equations in 1-dimensional domain reads

\[\begin{split} \partial_t \begin{bmatrix} \rho \\ m \\ E^t \end{bmatrix} + \partial_x \begin{bmatrix} m \\ \frac{m^2}{\rho} + p \\ \frac{m}{\rho} \left( E^t + p \right) \end{bmatrix} = \partial_x \begin{bmatrix} 0 \\ ( \mu + \lambda ) \partial_x \left( \frac{m}{\rho} \right) \\ ( \mu + \lambda ) \frac{m}{\rho} \partial_x \left( \frac{m}{\rho} \right) - q \end{bmatrix} \end{split}\]

The convective form using \((\rho, u, e)\) as primary variables reads

\[ \dots \]

Entropy function is chosen as the thermodynamic entropy volume density \(\eta = \rho s\). Entropy equation naturally follows from the thermodynamic relation \(ds = \frac{1}{T} \left( de - \frac{p}{\rho^2} d \rho \right)\),

\[\rho \partial_t s + \rho \mathbf{u} \cdot \nabla s = \frac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} + \frac{k |\nabla T|^2}{T^2} - \nabla \cdot \left(\frac{\mathbf{q}}{T}\right)\]

or in conservative form (using mass equation)

\[\partial_t \left( \rho s \right) + \nabla \cdot \left( \rho s \mathbf{u} \right) = \frac{2 \mu |\mathbb{D}|^2 + \lambda (\nabla \cdot \mathbf{u})^2}{T} + \frac{k |\nabla T|^2}{T^2} - \nabla \cdot \left(\frac{\mathbf{q}}{T}\right)\]

• Flux: \(F(\mathbf{u}) = \begin{bmatrix} m \\ \frac{m^2}{\rho} + p \\ \frac{m}{\rho}(E^t+p) \end{bmatrix} \)

• Entropy: \(\eta(\mathbf{u}) = \rho s \left( \rho, \mathbf{m}, E^t \right)\)

• Diffusion flux: \(\begin{bmatrix} 0 & 0 & 0 \\ - (\mu+\lambda) \frac{m}{\rho^2} & (\mu+\lambda) \frac{1}{\rho} & 0 \\ \dots & \dots & \dots \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ m \\ E^t \end{bmatrix}\)

Flux. First component

\[\begin{split}\nabla_{\mathbf{u}} F_1 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\end{split}\]

\[\nabla_{\mathbf{u} \mathbf{u}} F_1 = \mathbf{0}_3 \ .\]

Second component

\[\begin{split}\nabla_{\mathbf{u}} F_2 = \begin{bmatrix} - \frac{m^2}{\rho^2} + \Pi_{/\rho} \\ 2 \frac{m}{\rho} + \Pi_{/m} \\ \Pi_{E^t} \end{bmatrix}\end{split}\]

\[\begin{split}\nabla_{\mathbf{u} \mathbf{u}} F_2 = \begin{bmatrix} \dots \\ \dots \\ \dots \end{bmatrix} \ .\end{split}\]

Third component

\[\begin{split}\nabla_{\mathbf{u}} F_2 = \begin{bmatrix} \dots \\ \dots \\ \dots \end{bmatrix}\end{split}\]

\[\begin{split}\nabla_{\mathbf{u} \mathbf{u}} F_2 = \begin{bmatrix} \dots \\ \dots \\ \dots \end{bmatrix} \ .\end{split}\]

Entropy. Gradient

\[\begin{split}\nabla_{\mathbf{u}} \eta = \begin{bmatrix} s + \rho s_{/\rho} \\ \rho s_{/m} \\ \rho s_{/E^t} \end{bmatrix}\end{split}\]

and Hessian

\[\begin{split}\nabla_{\mathbf{u} \mathbf{u}} \eta = \begin{bmatrix} 2 s_{/\rho} + \rho s_{/\rho \rho} & s_{/m} + \rho s_{\rho m} & s_{/E^t} + \rho s_{\rho E^t} \\ \dots & \rho s_{m m} & \rho s_{m E^t} \\ \dots & \dots & \rho s_{E^t E^t} \\ \end{bmatrix} \end{split}\]

Compatibility of diffusion and entropy.

\[ \nabla_{\mathbf{u} \mathbf{u}} \eta \cdot \mathbf{D} = \mu \begin{bmatrix} \dots \end{bmatrix} \]

This matrix is semi-definite positive with one zero eigenvalue (associated with exact conservation and no dissipation in mass equation), and the other eigenvalue