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I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested in are:

Deterministic Cauchy problem: $\hspace{1cm} (1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)=u_0 (x) \end{cases} $

Deterministic Riemann problem: $\hspace{1cm} (2) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq0 \end{cases} \end{cases} $

Stochastic Riemann problem: $\hspace{1cm} (3) \hspace{1cm} \begin{cases} u_t+f(u)_x=g(u)W(t) \\[2ex] u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq0 \end{cases} \end{cases} $

Stochastic Cauchy problem: $\hspace{1cm} (4) \hspace{1cm} \begin{cases} u_t+f(u)_x=g(u)W(t) \\[2ex] u(x,0)=u_0 (x) \end{cases} $

Here u $\in \mathbb{R}^n$ and W(t) is a white noise. For $n=1$ we have one equation and for $n>1$ we have system of conservation laws. Function $g$ depends of $u$ so we say that we have multiplicative noise.

So far I only have found two references that study connection between solutions of problems (1) and (2). Those are:

  1. TP Liu, Large-Time Behavior of Solutions of Initial and Initial-Boundary Value Problems of a General System of Hyperbolic Conservation Laws, Comm. Math. Phys. Volume 55, Number 2 (1977), 163-177; projecteuclid, DOI: 10.1007/BF01626518.
  2. L. Hsiao, Quasilinear hyperbolic systems and dissipative mechanisms, World Scientific Publishing, 1997 - Chapter 5.

Anyone know any reference that connects any other pair of problems?

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