Distributional systems and entropy flux pairs

Let $d,m\in \mathbb{N}$, let $I\subseteq \mathbb{R}$ be an open time interval, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, and let $\mathcal{O}\subseteq {\mathbb{R}}^m$ be open and convex. Put $Q=I\times \mathrm{\Omega }$. A first-order conservation system on $\mathcal{O}$ consists of fluxes $F_j\in C^1(\mathcal{O};{\mathbb{R}}^m)$, $1\le j\le d$, and the equation ${\mathrm{\partial }}_{t}U+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}{F}_j(U)=0.$ A state field $U:Q\to \mathcal{O}$ is locally bounded if it is measurable and essentially bounded on every compact subset of $Q$. For an open set $Q$, $g\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(Q)$ means $g$ is measurable and ${\int }_K\mathrm{\mid }g\mathrm{\mid }<\mathrm{\infty }$ for every compact $K\subseteq Q$. Require also $U,F_1(U),\dots ,F_d(U)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(Q;{\mathbb{R}}^m)$. These fields then define regular distributions componentwise. The conservation system holds in distributions on $Q$ when, for every real-valued $\phi \in C_c^{\mathrm{\infty }}(Q;{\mathbb{R}}^m)$, ${\int }_Q(U\cdot {\mathrm{\partial }}_t\phi +{\sum }_{j=1}^d{F}_j(U)\cdot {\mathrm{\partial }}_{x_j}\phi )dxdt=0.$ This is the test-function meaning of the displayed derivative equation, using the distributional derivative convention in the supplied mathematics. The pairing is bilinear, without complex conjugation.

For an initial-value problem on $(0,T)\times \mathrm{\Omega }$, $T>0$, let $U_0:\mathrm{\Omega }\to \mathcal{O}$ be measurable and locally integrable. Require integrability of the state and fluxes up to the initial boundary: ${\int }_0^{\tau }\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_K(\mathrm{\mid }U\mathrm{\mid }+{\sum }_{j=1}^d\mathrm{\mid }{F}_j(U)\mathrm{\mid })dxdt<\mathrm{\infty }(0<\tau <T,K\subseteq \mathrm{\Omega }\text{ compact})\mathrm{.}$ This condition uses only positive-time values and does not assert a strong trace at $t=0$. A test function up to the initial boundary is the restriction of a smooth function defined on a neighborhood of $[0,T)\times \mathrm{\Omega }$, with compact support in $[0,T)\times \mathrm{\Omega }$; its value at $t=0$ need not vanish. The initial-value weak identity is ${\int }_0^T\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(U\cdot {\mathrm{\partial }}_t\phi +{\sum }_{j=1}^d{F}_j(U)\cdot {\mathrm{\partial }}_{x_j}\phi )dxdt+{\int }_{\mathrm{\Omega }}{U}_0(x)\cdot \phi (0,x)dx=0$ for every such real vector test function. The stated integrability up to the initial boundary makes every term finite, including for tests nonzero at $t=0$; a weak solution with initial data $U_0$ must satisfy these integrability conditions and this identity for all such tests.

A convex entropy-flux pair for the system is a function $\eta \in C^2(\mathcal{O};\mathbb{R})$ that is convex on the convex state domain $\mathcal{O}$ and functions $q_j\in C^1(\mathcal{O};\mathbb{R})$ satisfying $D{q}_j(V)=D\eta (V)D{F}_j(V)(V\in \mathcal{O},1\le j\le d),$ where convexity means $\eta ((1-\theta )V+\theta W)\le (1-\theta )\eta (V)+\theta \eta (W)$ for $V,W\in \mathcal{O}$ and $\theta \in [0,1]$, and $D\eta ,D{q}_j$ are row derivatives while $D{F}_j$ is the Jacobian matrix. For a state field $U$, require $\eta (U),q_1(U),\dots ,q_d(U)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ before using their distributions. The entropy inequality on $Q$ means ${\mathrm{\partial }}_t\eta (U)+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}{q}_j(U)\le 0,$ with the sign convention that this holds exactly when ${\int }_Q(\eta (U){\mathrm{\partial }}_t\psi +{\sum }_{j=1}^d{q}_j(U){\mathrm{\partial }}_{x_j}\psi )dxdt\ge 0$ for every nonnegative $\psi \in C_c^{\mathrm{\infty }}(Q)$. With initial data on $(0,T)\times \mathrm{\Omega }$, require also ${\int }_0^{\tau }\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_K(\mathrm{\mid }\eta (U)\mathrm{\mid }+{\sum }_{j=1}^d\mathrm{\mid }{q}_j(U)\mathrm{\mid })dxdt<\mathrm{\infty }(0<\tau <T,K\subseteq \mathrm{\Omega }\text{ compact})$ and $\eta (U_0)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$. Use every nonnegative scalar test up to the initial boundary defined above and add ${\int }_{\mathrm{\Omega }}\eta (U_0)\psi (0,x)dx$ to the entropy integral. These conditions make all terms finite even when the test is nonzero at $t=0$. A weak solution satisfying this inequality is entropy-admissible relative to that pair.

regular distributions from locally integrable fields

distributional derivative convention

local integrability and regular functional

For a state field $U\in C^1(Q;\mathcal{O})$ satisfying the conservation system pointwise on $Q$, the compatibility identity and the classical chain rule give ${\mathrm{\partial }}_t\eta (U)+{\sum }_j{\mathrm{\partial }}_{x_j}{q}_j(U)=D\eta (U)({\mathrm{\partial }}_{t}U+{\sum }_{j}D{F}_j(U){\mathrm{\partial }}_{x_j}U)=0$. For a merely weak solution this calculation is unavailable; the entropy inequality is an additional condition and does not follow from the weak conservation equation alone.

As a mathematical example of the notation, the one-dimensional spatial reduction of the compressible Euler model in three spatial dimensions uses $U=(\rho ,\rho v,E)$ and $F(U)=(\rho v,\rho v^2+p,v(E+p))$, with $E=\rho e+\frac{1}{2}\rho v^2$. Here $\rho >0$ is mass density in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $v$ is axial velocity in $\mathrm{m}{\mathrm{s}}^{-1}$, $e$ is specific internal energy in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, $E$ is total energy density in $\mathrm{J}{\mathrm{m}}^{-3}$, and $p=P(\rho ,e)$ is pressure in pascals, with $P$ a separately specified constitutive map. The components of $U$ have units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, and $\mathrm{J}{\mathrm{m}}^{-3}$; those of $F$ are the corresponding quantities multiplied by velocity. This is only a mathematical model form, not an adopted physical postulate. An entropy inequality does not establish general compressible-system existence or uniqueness.

classical chain rule