# Conservation laws and the heat equation

I studied equations of the form $$u_t+f(u)_x=0$$ are called conservation laws. Recently, somewhere I read that heat equation is a parabolic conservation law(I know that heat equation is a parabolic pde). I have the following doubts

1. Why heat equation is called a conservation law?

2. In general when do we say that a PDE is a conservation law?

(So far I thought equations of the form $$u_t+f(u)_x=0$$ are called conservation principle as they represent conservation in the integral form)

• Another common name for equations on this form is to say it's a continuity equation. – Winther May 1 '19 at 18:49

To answer this question, consider the equations in the form $$u_t+f(u)_x=0\quad x\in\Bbb R^n, n\ge 1\label{1}\tag{1}$$ which are usually called "Conservation laws", and let's do the simplifying assumption that \eqref{1} is a scalar conservation law, i.e. $$u$$ is a real valued function. Then, a more comfortable way to write \eqref{1} is the following: $$u_t+\operatorname{div}\mathbf{f}(u)=0\label{1'}\tag{1'}$$ where $$\mathbf{f}(u)=\mathbf{f}\big(u(x,t)\big)=\left. \begin{pmatrix} f\big(u(x,t)\big)\\ f\big(u(x,t)\big)\\ \vdots\\ f\big(u(x,t)\big) \end{pmatrix}\quad\right\}\text{ n rows}.$$ Now consider a domain $$V\in\Bbb R^n$$ bounded by an hypersurface $$\partial V$$, integrate the two sides of the equation over it and apply the divergence (Gauss-Green) theorem: we have $$\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_V u(x,t)\,\mathrm{d} x + \int\limits_{\partial V} \mathbf{f}\big(u(x,t)\big)\cdot\nu_x\,\mathrm{d}\sigma_x = 0 \label{2}\tag{2}$$ Thus the variation of the total quantity $$u$$ with respect to the time $$t$$ on the domain $$V$$ equals the flux of the vector $$\mathbf{f}(u)$$ across the boundary of the domain: there is no annihilation nor generation of the quantity $$u$$ inside $$V$$, or in simple words, the available total quantity of $$u$$ is conserved.
In sum, every evolution equation of the form $$\mathbf{u}_t(x,t)+\operatorname{div}\mathbf{A}\big(x,t,\mathbf{u}(x,t)\big)=0\quad x\in\Bbb R^n,\mathbf{u}\in\Bbb R^m, \mathbf{A}\in M^{m\times n}(\Bbb R)\label{3}\tag{3}$$ is a conservation law since its integral form is totally analogous to \eqref{2} and gives rise to the same interpretation.
The heat equation has the form \eqref{1'} since $$u_t-\Delta u=0\iff u_t-\operatorname{div}\cdot\operatorname{grad} u=0,$$ therefore it give rise to a formula of type \eqref{2} with $$\mathbf{f}(u)= -\operatorname{grad}u$$ so it express, by definition, a (very particular) scalar conservation law.