| Abstract: |
| We study generalized time-fractional stochastic conservation laws, the prototype being of the form
\[ d g_{1-\alpha} \ast(u-u_0) + \mbox{ div}f(u)dt = I^{1-\beta}h dW,\]
where $g_{1-\alpha}(t) = \frac{t^{-\alpha}}{\Gamma(1- \alpha)}$ ($\alpha \in (0,1)$), i.e.,
\[ \partial_t g_{1 - \alpha} \ast (u-u_0)= \partial_t^\alpha (u-u_0)\]
is the fractional time-derivative in the sense of Riemann-Liouville, $f: \mathbb{R} \rightarrow \mathbb{R}^N$ is a smooth function, $I^{1- \beta}$ is the fractional integral of order $1 - \beta$ in the sense of Riemann-Liouville ($\beta=1$ corresponds to the classical additive stochastic noise $hdW$ with a given function $h=h(\omega, t,x)$) and $W=(W_t,, \mathcal{F}_t; 0 \leq t \leq T)$ is one-dimensional Brownian motion.\
Under appropriate assumptions on $\alpha$ and $\beta$ we prove existence and uniqueness of a stochastic entropy solution for arbitrary $L^2$-initial data.\
An interesting open question is whether it is possible to generalize these results to the case of a multiplicative stochastic noise. The main difficulty is that an It\^o type formula is not known to hold in the time-fractional derivative case. |
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