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Bounds on front speeds for inviscid and viscous $$G$$-equations. (English) Zbl 1256.35090
Summary: $$G$$-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
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