Communications in Mathematical Physics

, Volume 359, Issue 2, pp 515–533 | Cite as

Curvature Effect in Shear Flow: Slowdown of Turbulent Flame Speeds with Markstein Number

  • Jiancheng Lyu
  • Jack Xin
  • Yifeng Yu


It is well-known in the combustion community that curvature effect in general slows down flame propagation speeds because it smooths out wrinkled flames. However, such a folklore has never been justified rigorously. In this paper, as the first theoretical result in this direction, we prove that the turbulent flame speed (an effective burning velocity) is decreasing with respect to the curvature diffusivity (Markstein number) for shear flows in the well-known G-equation model. Our proof involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. On a related fundamental issue, we solve the selection problem of weak solutions or find the “physical fluctuations” when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA

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