hyperbolic-equationspartial differential equations
Consider the IVP
\begin{eqnarray}
u_t+F(x,u)_x=S(x,u)\\
u(x,0)=u_0(x)
\end{eqnarray}
If $S(x,u)=0$ and $u\in C([0,T],L^1(\mathbb{R})),$ then we have $$\int\limits_{\mathbb{R}}u_0(x)dx=\int\limits_{\mathbb{R}}u(x,t)dx.$$ (physically which can be interpreted as conservation of mass…)
What happens when $S(x,u)\neq 0$
P.S. Please give a proof or suggest a reference..
The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = \frac{1}{2}u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.
In the case where the flux $f$ is not convex, these are the possible types of waves:
A rather practical method of solving such problems is convex hull construction: [1]
The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $\lbrace (u, y) : u_R ≤ u ≤ u_L \text{ and } y ≤ f (u)\rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $\lbrace (u, y) : u_L ≤ u ≤ u_R \text{ and } y ≥ f (u)\rbrace$.
Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.
One can also use Osher's expression of general similarity solutions $u(x,t) = v(\xi)$, which writes [1]
$$ v(\xi) = \left\lbrace \begin{aligned} &\underset{u_L\leq u\leq u_R}{\text{argmin}} \left(f(u) - \xi u\right) && \text{if}\quad u_L\leq u_R \, ,\\ &\underset{u_R\leq u\leq u_L}{\text{argmax}} \left(f(u) - \xi u\right) && \text{if}\quad u_R\leq u_L \, . \end{aligned} \right. $$
To summarize, here are the different entropy solutions and their validity in the case $f(u) = \frac{1}{3}u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^{-1}(\xi) = \pm\sqrt{\xi}$. Using the convex hull construction method, one gets:
(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.
The energy of the second-order wave equation $u_{tt} = u_{xx}$ is the sum of the total kinetic energy and of the total potential energy: $E = \frac12 \int_{\Bbb R} (u_t)^2 + (u_x)^2\, \text d x$. For smooth solutions, we have $$ \begin{aligned} \frac{\text d}{\text dt} E &= \int_{\Bbb R} u_t u_{tt} + u_x u_{xt}\, \text d x \\ &= \int_{\Bbb R} u_t u_{xx} + u_x u_{xt}\, \text d x \\ &= \left.[u_t u_{x}]\right|_{x \in \Bbb R} + \int_{\Bbb R} u_{tx} u_{x}\, \text d x - \int_{\Bbb R} u_x u_{xt}\, \text d x \\ &= 0 \end{aligned} $$ by integration by parts, if $u$, $u_t$ and $u_x$ have compact support.
One notes that we have the following factorization of the d'Alembert operator $$ \square u = (\partial_{tt} - \partial_{xx}) u = (\partial_{t} - \partial_{x})(\partial_{t} + \partial_{x}) u \, . $$ Hence, we may think that $\frac{\text d}{\text dt} E = 0$ remains true for the scalar advection equation $u_t \pm u_x = 0$. Indeed, the same derivation as for the second-order wave equation can be made due to the fact that $u_{tt} = \mp u_{tx} = u_{xx}$. Moreover, with $\Psi = \frac12 \int_{\Bbb R} u^2 \, \text d x$, we have $$ \begin{aligned} \frac{\text d}{\text dt} \Psi &= \int_{\Bbb R} u u_t \, \text d x \\ &= \mp \int_{\Bbb R} u u_x \, \text d x \\ &= \mp \tfrac12 \left.[u^2]\right|_{x \in \Bbb R} \\ &= 0 \, , \end{aligned} $$ for smooth $u$ with compact support. In the case of the conservation law $u_t + f(u)_x = 0$, we still have $$ \begin{aligned} \frac{\text d}{\text dt} \Psi &= \int_{\Bbb R} u u_t \, \text d x \\ &= - \int_{\Bbb R} u f(u)_x \, \text d x \\ &= -\left.[u f(u)]\right|_{x \in \Bbb R} + \int_{\Bbb R} u_x f(u) \, \text d x \\ &= -\left.[u f(u)]\right|_{x \in \Bbb R} + \left.[f(u)]\right|_{x \in \Bbb R} \\ &= 0 \end{aligned} $$ by integration by parts, if $u$ and $f\circ u$ have compact support. For instance, it should work for $u$ with compact support and for a flux $f$ which vanishes at the origin --- without loss of generality, as $g:u\mapsto f(u) - f(0)$ satisfies $u_t + g(u)_x = 0$ and $g(0) = 0$. Note that the spatial density $\psi = \frac12 u^2$ of $\Psi = \int \psi\,\text d x$ is a convex function of $u$. Hence, $\psi$ is a mathematical entropy, in the sense that $\psi(u)_t + h(u)_x = 0$, for the corresponding entropy flux $ h(u) = \int^u \psi'(v) f'(v)\,\text d v $.
All these estimates can be used for the uniqueness proof.
Best Answer
If $S \neq 0$ then $S$ serves it's role as a source, for example, something that can "introduce mass". Let's compute as before: \begin{align*} \partial_t \int_{\mathbb{R}} u(x, t)\, dx &= \int_{\mathbb{R}} u_t(x, t)\, dx \\ &= -\int_{\mathbb{R}} \frac{d}{dx}F(x, u(x, t))\, dx + \int_{\mathbb{R}} S(x, u(x, t))\, dx \\ &= \int_{\mathbb{R}}S(x, u(x, t))\, dx. \end{align*} The $F$ integral vanishes if we assume sufficient decay, e.g. $F$ goes to zero at $\pm \infty$, by the fundamental theorem of calculus. Then we have $$ \text{Mass at time $t$ = }\int_{\mathbb{R}}u(x, t)\, dx = \int_{\mathbb{R}}u_0(x)\, dx + \int_0^t \int_{\mathbb{R}}S(x, u(x, s))\, dx \, ds. $$ So the interpretation is that the quantity $\int_{\mathbb{R}}u(x, t)\, dx$, rather than being constant in $t$, changes from its initial value by integrating the source $S$.