Dynamics and universality of an isothermal combustion problem in 2D

    Authors

    Y. W. Qi

    Comments

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    Abbreviated Journal Title

    Rev. Math. Phys.

    Keywords

    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

    Abstract

    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 [7]), 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 [6], Bricmont, Kupiainen and Xin, [7], (see also [9]) and key estimates using the PDE method.

    Journal Title

    Reviews in Mathematical Physics

    Volume

    18

    Issue/Number

    3

    Publication Date

    1-1-2006

    Document Type

    Review

    Language

    English

    First Page

    285

    Last Page

    310

    WOS Identifier

    WOS:000238508000003

    ISSN

    0129-055X

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