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

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 (p) = |p| j for j ≥ 2 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 (x, t) 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.for constants a i j , {µ i }, we say that (X 1 , ..., X n ) form a multivariate normal distribution. We also define the covariance of two random variables below.Definition 2.2. Let X i and X j be random variables. We define the covariance of the random variables byThis is the basis of the definition for a Gaussian stochastic process. In particular, one has the following [27]:

Regularizing effect for conservation laws with a Lipschitz convex flux

“…Otherwise, it is defined a.e in [4]. To be self contained, Proposition 4.1 has been added here with a short proof (see also [5,6]).…”

“…Taking the limit n → ∞, thanks to Lemma 4.1, u has an explicit continuous formula. Thus, u is a weak entropy solution satisfying ( 5) and (6). Noting that b(ξ) = u l ∀ξ ∈ [a − (u l ), a + (u l )], ā(u l ) in ( 26) can be replaced by a − (u l ) or by a + (u l ).…”