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A survey of results on conservation laws with deterministic and random initial data

The author thanks Professors Menon and Dafermos and Dr. Kaspar for valuable discussions. This work was partially supported by NSF grants DMS 1411278 and DMS 1148284 as well as the NSF Graduate Research Fellowship.
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  • This expository paper examines key results on the dynamics of nonlinear conservation laws with random initial data and applies some theorems to physically important situations. Conservation laws with some nonlinearity, e.g. Burgers' equation, exhibit discontinuous behavior, or shocks, even for smooth initial data. The introduction of randomness in any of several forms into the initial condition renders the analysis extremely complex. Standard methods for tracking a multitude of shock collisions are difficult to implement, suggesting other methods may be needed. We review several perspectives into obtaining the statistics of resulting states and shocks. We present a spectrum of results from a number of works, both deterministic and random. Some of the deep theorems are applied to important discrete examples where the results can be understood in a clearer, more physical context.

    Mathematics Subject Classification: Primary: 35F50, 35F55; Secondary: 35L65.


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  • Figure 1.  (a) By taking a cross-section in time, one can obtain a cumulative distribution function of the mass as a function of position; (b) Illustration of the potential as a function of mass; (c) Illustration of the flux function of mass

    Figure 2.  (a) Cumulative distribution function of mass as a function of position; (b) Construction of $\Psi\left( 0, x\right) $

    Figure 3.  (a) Representation of mass in cumulative distribution form up to a point $x$ in the $xt$ plane; (b)-(d) Plot of the expression $\Phi^{0}\left( m\right) +tA\left( m\right) $ for times $t = 0, 1, 2$ respectively in solid lines; following the dashed lines forms the convex hull, yielding the Legendre transform $\Phi_{n}\left( t, m\right) .$ Note that for (b) and (c), the expression and its convex hull are identical, and in (d) there is a distinction, with the convex hull indicated by the dashed blue line

    Figure 4.  Evolution of the discrete example and mapping back using the flow map. Highlighted in blue (long-short dash lines) are intervals unchanged under the flow map. In red (long dashed line) are intervals for which the flow map inverse is undefined. The points in green correspond to single points for which an entire interval is mapped back onto, which occurs in notably many cases. For example, $\varphi_{t_{2}^{\ast}}^{-1}\left( I_{2}^{1\ast}\right) = \left\{ 0\right\} $ and $\varphi_{t_{1}^{\ast}}^{-1}\left( \left\{ -2\right\} \right) = \left\{ \emptyset\right\} $

    Figure 5.  Graphs of (a) $\int_{0+0}^{y-0}tu_{0}\left( \eta\right) dm_{0}\left( \eta\right) $ (with $t = 1$), (b) $\int_{0+0}^{y-0}\left( \eta-x\right) dm_{0}\left( \eta\right) $, (c) $F\left( y;0, 1\right) $

    Figure 6.  For a shock at a point $x^{\ast}$, we slide the parabola $\left( x-x^{\ast}\right) ^{2}/2$ down until we have (at least) two contact points with the Brownian path, but in such a way that the parabola does not cross the Brownian path. If there are more than two, we consider only the first and last contact points. These points are given by $\left( \xi_{-}, \left( \xi _{-}-x^{\ast}\right) ^{2}/2\right) $ and $\left( \xi_{+}, \left( \xi _{+}-x^{\ast}\right) ^{2}/2\right) $. The shock is then described by the parameters $\mu = \xi_{+}-\xi_{-}$ and $\nu = x^{\ast}-\xi_{-}.$ This figure is based off Figure 1, [16]

    Figure 7.  Illustration of the flux function as described above

    Figure 8.  (a)-(b) Construction of the test functions $\varphi_{k}\left( u\right) $ and $\psi_{k}\left( u\right) $; (c) Illustration of a shock, with positive contribution from $\partial_{t}p_{1}\left( x, t;u_{l}\right) $ (upward arrow, blue), and negative contribution from $p_{2}\left( x, x+, t;u_{l}, u_{r}\right) $ (right arrow, red)

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