Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

Abstract

In this paper we study the long time dynamics of the solutions to the initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its asymptotic stability, we focus our attention on the phenomenon of 'metastability', whereby the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches a steady state in a very long time interval. Numerical simulations illustrate the results.

Figures (9)

• Figure 1: The solution to $\partial_t u = \varepsilon \partial_x^2 u -u\partial_x u$ with $\varepsilon=0.07$ and initial datum $u_0(x)= \frac{1}{2}x^2-x-\frac{1}{2}$ in grey. The motion of the time-dependent solution towards its asymptotic configuration, given by the hyperbolic tangent centered in zero, takes place in an exponentially long time interval, and a metastable behavior is observed.
• Figure 2: The dynamics of the solution to \ref{['burgers2']} for $\varepsilon=0.005$, $f(u)=u^2/2$ and $u_0$ increasing with $u_\pm=\pm\sqrt{\varepsilon}$ (left) and decreasing with $u_\pm=\mp\sqrt{\varepsilon}$ (right). In both pictures, an interface is formed in a short time. However, in the left picture no metastable behavior is observed as one can see that, for $t=100$, the solution has already reached the steady state. On the opposite, in the right-hand picture the interface is still very far from zero for times of the same order (the plots for $t=50$ and $t=100$ are indistinguishable).
• Figure 3: The dynamics of the solution to \ref{['burgers2']} for $\varepsilon=0.005$, $f(u)=u^2/2$ and two different non-monotone initial data (plotted in grey) connecting the values $u_\pm=\mp\sqrt{\varepsilon}$. As we can see, it is neither necessary for the initial datum to be decreasing nor to be such that $\|u_0\|_{{}_{L^\infty}} \leq |u_\pm|$ to observe a metastable behavior.
• Figure 4: The dynamics of the solution to \ref{['burgers2']} for $\varepsilon=0.005$, $f(u)=u^2/2$, $u_\pm=\mp\sqrt\varepsilon$ and $u_0$ decreasing and such that $u_0(x_0)=0$ for some $x_0>0$. In this case, the interface is moving with negative speed.
• Figure 5: The dynamics of the solution to \ref{['burgers2']} with $u_\pm = \mp \sqrt{\varepsilon}$, initial datum $u_0(x)=\sqrt{\varepsilon}\left(\frac{1}{2}x^2-x-\frac{1}{2}\right)$ and $\varepsilon=0.005$; the flux function is $f(u)=\frac{1}{2}(u+a \, \sqrt{\varepsilon})^2$ with $a=0.25$ (left) and $a=0.1$ (right), so that $f(u_+)\neq f(u_-)$. We can see that the asymptotic steady state is attained in a short time scale. We also observe that $f(u_-)- f(u_+)= 2 a \varepsilon$ and the smaller this difference, the slower the convergence.

Theorems & Definitions (24)

• Theorem \oldthetheorem
• Theorem \oldthetheorem
• proof : Proof of Theorem \ref{['teoincr']}
• Theorem \oldthetheorem
• Example \oldthetheorem
• Theorem \oldthetheorem
• Proposition \oldthetheorem
• proof : Proof of Proposition \ref{['prop:aprioriux']}
• Remark \oldthetheorem
• Proposition \oldthetheorem