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The viscous entropy dissipation identity

Statement

Let $n \geq 1$ , $ε > 0$ , $f \in C^{1} (R; R^{n})$ and let $u^{ε} \in C^{1, 2} (Π_{T})$ be a classical solution of the viscous conservation law $u_{t}^{ε} + {div}_{x} f (u^{ε}) = ε Δ u^{ε} in Π_{T}$ (Scalar conservation laws, fluxes and Cauchy data, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field). For every convex $η \in C^{2} (R)$ and entropy flux $q$ with $q^{'} (s) = η^{'} (s) f^{'} (s)$ , the pointwise viscous entropy balance is $η (u^{ε})_{t} + {div}_{x} q (u^{ε}) = ε Δ_{x} η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2} \leq ε Δ_{x} η (u^{ε}) .$ The nonpositive term is the entropy dissipation; for fixed $ε > 0$ this is a balance with diffusion, not the first-order entropy inequality (Convex entropy--entropy flux pairs, Divergence and curl of a $C^{1}$ vector field).

Facts & Assumptions

Given: $n \geq 1$ , $ε > 0$ , $f \in C^{1} (R; R^{n})$ , a classical solution $u^{ε} \in C^{1, 2} (Π_{T})$ of the viscous conservation law, and a convex $η \in C^{2} (R)$ with entropy flux $q$ , $q^{'} = η^{'} f^{'}$ .

[F1]

Chain rules for a $C^{1}$ function of a $C^{1, 2}$ function: the composition $η (u^{ε})$ is $C^{1, 2}$ , while $q (u^{ε})$ and $f (u^{ε})$ are $C^{1}$ , with $\partial_{t} η (u^{ε}) = η^{'} (u^{ε}) u_{t}^{ε}$ , and ${div}_{x} q (u^{ε}) = q^{'} (u^{ε}) \cdot \nabla_{x} u^{ε} = η^{'} (u^{ε}) f^{'} (u^{ε}) \cdot \nabla_{x} u^{ε}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Convex entropy--entropy flux pairs).

[F2]

Laplacian of a composition: $Δ_{x} η (u^{ε}) = η^{'} (u^{ε}) Δ u^{ε} + η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ , obtained by applying the chain rule and the product rule to the components $\partial_{i} (η (u^{ε})) = η^{'} (u^{ε}) \partial_{i} u^{ε}$ and summing in $i$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

technique · direct

1.1F1F2

By [F1] the two left-hand terms are $η (u^{ε})_{t} = η^{'} (u^{ε}) u_{t}^{ε}$ and ${div}_{x} q (u^{ε}) = η^{'} (u^{ε}) f^{'} (u^{ε}) \cdot \nabla_{x} u^{ε}$ , and by [F2] the diffusion term is $Δ_{x} η (u^{ε}) = η^{'} (u^{ε}) Δ u^{ε} + η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ .

2.1step 1.1F1algebra

Multiplying the viscous equation pointwise by $η^{'} (u^{ε})$ and adding the second identity of step 1.1 gives $η (u^{ε})_{t} + {div}_{x} q (u^{ε}) = η^{'} (u^{ε}) (u_{t}^{ε} + {div}_{x} f (u^{ε}) - ε Δ u^{ε}) + ε Δ_{x} η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2} = ε Δ_{x} η (u^{ε}) - ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ , which is the asserted balance.

3.1step 2.1F2∎

The convexity assumption gives $η^{''} \geq 0$ , so the dissipation term $- ε η^{''} (u^{ε}) ∣ \nabla_{x} u^{ε} ∣^{2}$ is nonpositive pointwise and the balance implies the stated inequality; the term $ε Δ_{x} η (u^{ε})$ may change sign and cannot be dropped pointwise for fixed $ε > 0$ .

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Sources

S. N. Kruzhkov, “First order quasilinear equations in several independent variables,” Mat. USSR-Sbornik 10 (1970), 217–243, complete English translation (standard reference, not scraped)
G. A. Chechkin and A. Yu. Goritsky (translated by B. Andreianov), “S. N. Kruzhkov’s lectures on first-order quasilinear PDEs,” in Analytical and Numerical Aspects of PDEs, de Gruyter 2009, complete lecture-notes text (standard reference, not scraped)