Bounds on front speeds for inviscid and viscous \(G\)-equations.

*(English)*Zbl 1256.35090Summary: \(G\)-equations are well-known front propagation models in combustion
and describe the front motion law in the form of local normal velocity
equal to a constant (laminar speed) plus the normal projection
of fluid velocity. In level set formulation, \(G\)-equations are Hamilton-
Jacobi equations with convex (\(L^1\) type) but non-coercive Hamiltonians.
We study front speeds of both inviscid and viscous \(G\)-equations
in mean zero flows, and compare the qualitative speed properties with
those of quadratically nonlinear Hamilton-Jacobi equations and KPP
(Kolmogorov-Petrovsky-Piskunov) fronts (with minimal speed). For
the inviscid case, we analyze a variational solution formula (control
representation) by choosing suitable test functions. For the viscous
case, we analyze traveling front equations which agree with the cell
problem of homogenization. We found that viscosity can drastically
alter the front speed growth law of \(G\)-equations. Without viscosity,
front speed grows like \(O(A/\log A)\) in cellular flows of large amplitude
\(A\). With proper viscosity, the front speed grows no faster than
\(O(\sqrt{\log A})\). In contrast, the KPP front speed grows like \(O(A^{1/4})\) in general cellular flows at any fixed viscosity. The \(L^1\) type nonlinearity
appearing in the \(G\)-equation makes the key difference.

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35F21 | Hamilton-Jacobi equations |

35F25 | Initial value problems for nonlinear first-order PDEs |

76M30 | Variational methods applied to problems in fluid mechanics |

76N25 | Flow control and optimization for compressible fluids and gas dynamics |