Integrable weak continuum balance and entropy adoption

Adopt the mathematical Countable Choice convention: every sequence of nonempty sets admits a choice function. Use the complete Euclidean Lebesgue spaces, measurable maps, almost-everywhere equality and Lebesgue integrals of the conservative weak-form definition. This convention specifies the mathematical weak branch.

Work in a classical Cartesian inertial frame with SI units on a fixed open spatial region $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ and time interval $(0,T_f)$, where $T_f>0$ is measured in seconds. Write $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ for functions whose absolute value is integrable on every compact subset. Let all fields be measurable functions of $(t,x)$ on $\mathrm{\Omega }\times (0,T_f)$. Space-time integrals use the restriction of ${\lambda }_4$ to $\mathrm{\Omega }\times [0,T_f)$, and initial integrals use the restriction of ${\lambda }_3$ to $\mathrm{\Omega }$; both restrictions use their trace sigma-algebras. The displayed iterated notation uses the completed-product convention, with section integrals for almost every time, as in the conservative weak-form definition. Require $T>0$ almost everywhere wherever $q\mathrm{/}T$ or $r\mathrm{/}T$ is used. Let $\rho \ge 0$ be mass density in kilograms per cubic metre, $u$ velocity in metres per second, $\sigma$ Cauchy stress in pascals, $b$ body acceleration in metres per second squared, $e$ specific internal energy in joules per kilogram, $q$ outward-positive conductive heat flux in watts per square metre, $r$ specific heat supply in watts per kilogram, $T>0$ absolute temperature in kelvins, and $s$ specific entropy in joules per kilogram kelvin. Set $E:=e+\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}2$, use $t=\sigma n$ for traction and rowwise matrix divergence, and define the test-gradient by $(\mathrm{\nabla }\psi {)}_{ij}:={\mathrm{\partial }}_{x_j}{\psi }_i$ and the Frobenius product by $A:B:={\sum }_{i,j}{A}_{ij}{B}_{ij}$.

A test function is a $C^1$ function on $\mathrm{\Omega }\times [0,T_f)$, compactly supported in space-time and vanishing near $T_f$; it may be nonzero at $t=0$. Let scalar test functions be $\varphi ,\eta ,\zeta$ and vector test function be $\psi$. Adopt as a separate weak-branch model assumption that the displayed conservative variables, fluxes and sources below belong to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ and satisfy the following identities for every indicated test function. Initial conservative data ${\rho }_0,m_0,{\mathcal{E}}_0,S_0\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ represent initial mass, momentum, total-energy and entropy densities; the initial terms specify distributional data and do not assert a strong trace.

Mass balance, for every scalar $\varphi$,

${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho {\mathrm{\partial }}_t\varphi +\rho u\cdot \mathrm{\nabla }\varphi )dxdt+{\int }_{\mathrm{\Omega }}{\rho }_0\varphi (0,x)dx=0.$

Momentum balance, for every vector $\psi$,

${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho u\cdot {\mathrm{\partial }}_t\psi +(\rho u\otimes u-\sigma ):\mathrm{\nabla }\psi +\rho b\cdot \psi )dxdt+{\int }_{\mathrm{\Omega }}{m}_0\cdot \psi (0,x)dx=0.$

Total-energy balance, for every scalar $\eta$,

${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho E{\mathrm{\partial }}_t\eta +(\rho Eu-{\sigma }^{\mathsf{T}}u+q)\cdot \mathrm{\nabla }\eta +(\rho b\cdot u+\rho r)\eta )dxdt+{\int }_{\mathrm{\Omega }}{\mathcal{E}}_0\eta (0,x)dx=0.$

Physical entropy inequality, for every nonnegative scalar $\zeta \ge 0$,

${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho s{\mathrm{\partial }}_t\zeta +(\rho su+q\mathrm{/}T)\cdot \mathrm{\nabla }\zeta +(\rho r\mathrm{/}T)\zeta )dxdt+{\int }_{\mathrm{\Omega }}{S}_0\zeta (0,x)dx\le 0.$

Every conservative density, product, flux and source displayed in these tests is required to be locally integrable up to $t=0$, including $\rho u\otimes u$, $\rho Eu$, ${\sigma }^{\mathsf{T}}u$, $q\mathrm{/}T$ and $\rho r\mathrm{/}T$: its absolute value has finite integral on every compact subset of $\mathrm{\Omega }\times [0,T_f)$ in the relative topology, using its values for $t>0$. This condition assigns no point values at $t=0$ and asserts no strong initial trace. Interior test functions impose no boundary condition; walls, thermal exchange, periodic identifications and stronger initial trace conditions are separate data. This weak mass/momentum/energy adoption and the physical entropy inequality are not inferred for arbitrary discontinuous or vacuum states from the smooth balance postulate. The source term and the sign of the entropy inequality use the outward-positive heat-flux convention above.

Locally integrable weak equations, test functions and initial terms

Smooth-branch adopted continuum balance model

Powers, Lecture Notes on Intermediate Fluid Mechanics, §§4.4–4.8 and §9.4.1

Tong, Fluid Mechanics, §3.1.2