# zbMATH — the first resource for mathematics

Homogenization of the $$G$$-equation with incompressible random drift in two dimensions. (English) Zbl 1241.35021
The authors investigate the asymptotic behavior as $$\varepsilon\rightarrow 0$$ of the Hamilton-Jacobi equation $u_t^\varepsilon+V(x/\varepsilon,\omega)\cdot Du^\varepsilon=|Du^\varepsilon|,$ for $$t>0$$, $$x\in\mathbb R^2$$, together with the initial condition $$u^\varepsilon=u_0(x)$$ for $$t=0$$, $$x\in\mathbb R^2$$. The random vector field $$V$$ is assumed to be statistically stationary and ergodic, while the initial condition $$u_0$$ is assumed to be bounded and uniformly continuous. This is a model for flame propagation in a turbulent fluid in the regime of thin flames, the level sets $$u^\varepsilon$$ represent the flame surface, and $$V$$ describes the velocity of the underlying fluid.
In order to perform a stochastic homogenization of the $$G$$-equation, it is assumed that $$V$$ is almost surely of class $$C^1$$, essentially bounded and divergence free, i.e. $$\nabla\cdot V(x,\omega)=0$$ for all $$x\in\mathbb R^2$$. Then, with probability one, the solution $$u^\varepsilon$$ converges as $$\varepsilon\rightarrow0$$ locally uniformly to the solution $$\bar u$$ of the deterministic problem $\bar u_t = \bar H(D\bar u),\quad t>0,x\in\mathbb R^2,$ with the same initial condition $$\bar u(0,x)=u_0(x)$$, where the function $$\bar H:\mathbb R^2\rightarrow[0,\infty)$$ is convex and homogeneous of degree one.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35F21 Hamilton-Jacobi equations 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: