Dynamics and universality of an isothermal combustion problem in 2D
Abbreviated Journal Title
Rev. Math. Phys.
auto-catalytic chemical reactions; critical nonlinearity; anomalous; exponent; renormalization group; universal spatial-temporal profiles; UNEQUAL DIFFUSION RATES; RENORMALIZATION-GROUP; CRITICAL NONLINEARITY; CUBIC AUTOCATALYSIS; TRAVELING WAVES; WEAK SOLUTIONS; ASYMPTOTICS; EQUATIONS; SYSTEMS; FLAMES; Physics, Mathematical
In this paper, the Cauchy problem of the system u(1,t) = Delta(u1) - u(1)u(2)(m), u(2,t) = d Delta u(2) + u(1)u(2)(m) is studied, where x is an element of R-2, m >= 1 and d > 0 is the Lewis number. This system models isothermal combustion (see ), and auto-catalytic chemical reaction. We show the global existence and regularity of solutions with non-negative initial values having mild decay as vertical bar x vertical bar -> infinity. More importantly, we establish the exact spatio-temporal profiles for such solutions. In particular, we prove that for m = 1, the exact large time behavior of solutions is characterized by a universal, non-Gaussian spatio-temporal profile, with anomalous exponents, due to the fact that quadratic nonlinearity is critical in 2D. Our approach is a combination of iteration using Renormalization Group method, which has been developed into a very powerful tool in the study of nonlinear PDEs largely by the pioneering works of Bricmont, Kupiainen and Lin , Bricmont, Kupiainen and Xin, , (see also ) and key estimates using the PDE method.
Reviews in Mathematical Physics
"Dynamics and universality of an isothermal combustion problem in 2D" (2006). Faculty Bibliography 2000s. 7928.