Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics

Within the framework of space-time correlated continuous-time random walk model, anomalous diffusion of particle moving in the velocity field is studied in this paper. The weak asymptotic form ω(t) ∼ t−(1+α); 1 < α < 2 for large t, is considered to be the waiting time distribution. Analytical results reveal that the diffusion in the velocity field, i.e., the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. Numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can make the diffusion to be characterized by diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. Especially, for the waiting time displaying weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescale. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful for the relevant experimental research.

Uniform in Time Description for Weak Solutions of the Hopf Equation with Nonconvex Nonlinearity

We consider the Riemann problem for the Hopf equation with concave-convex flux functions. Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition.