A minimization approach to conservation laws with random initialconditions and non-smooth, non-strictly convex flux

Carey Caginalp; Carey Caginalp

doi:10.3934/Math.2018.1.148

AIMS Mathematics

2018, Volume 3, Issue 1: 148-182. doi: 10.3934/Math.2018.1.148

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Research article

A minimization approach to conservation laws with random initialconditions and non-smooth, non-strictly convex flux

Carey Caginalp ^,

University of Pittsburgh, Mathematics Department, 301 Thackeray Hall, Pittsburgh PA 15260

Received: 19 February 2018 Accepted: 11 March 2018 Published: 19 March 2018

We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) = \left\vert p\right\vert .{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum have some special properties. We calculate the transition probabilities between various cases and examine the variance of the solution $w\left(x, t\right)$ in both $x$ and $t$. We also describe how results may be obtained for a non-discretized version of a Gaussian stochastic process by taking the continuum limit as the partition becomes more fine.
- conservation laws,
- random initial conditions,
- Lax-Oleinik,
- shocks,
- variational problemsin differential equations,
- minimization,
- stochastic processes
Citation: Carey Caginalp. A minimization approach to conservation laws with random initialconditions and non-smooth, non-strictly convex flux[J]. AIMS Mathematics, 2018, 3(1): 148-182. doi: 10.3934/Math.2018.1.148

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Abstract

We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) = \left\vert p\right\vert .{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum have some special properties. We calculate the transition probabilities between various cases and examine the variance of the solution $w\left(x, t\right)$ in both $x$ and $t$. We also describe how results may be obtained for a non-discretized version of a Gaussian stochastic process by taking the continuum limit as the partition becomes more fine.

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