Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hlder regularity (dimension three) of the shear flows. Remarks are made on the quadratic and linear laws of front speed expectation in the small and large root mean square regimes.

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L p-L q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n>= 3, the global existence and decay rates of solutions to the Cauchy problem are

G-equations are well-known front propagation models in turbulent combustion which 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 HamiltonJacobi equations with convex (<i>L</i> ¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $${\bar H}$$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed <i>s</i> _T. An important problem in turbulent combustion theory is to study properties of <i>s</i> _T, in particular how <i>s</i> _T depends on the flow amplitude <i>A</i>. In this paper, we study the behavior of $${\bar H=\bar H(A,d)}$$ as <i>A</i> + at any fixed diffusion constant <i>d</i> &gt; 0. For cellular flow, we show that$$\bar H(A,d)\leqq

We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier--Stokes--Fourier system. The method is based on the relative entropy method and does not assume any smallness conditions on the discontinuity nor on the bounded variation norm of the initial data. It is proved that for any viscosity , the solution of the compressible Navier--Stokes--Fourier system (with well-prepared initial value) converges, when the heat conductivity tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate . It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solution for a system.

ABSTRACT We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no $H$-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent $-\frac23$ for small size to $-\frac32$ for large size, with an exponential cut-off.