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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.

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)
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