Title

Dynamics And Universality Of An Isothermal Combustion Problem In 2D

Keywords

Anomalous exponent; Auto-catalytic chemical reactions; Critical nonlinearity; Renormalization group; Universal spatial-temporal profiles

Abstract

In this paper, the Cauchy problem of the system u1,t = Δu1 - u1u2m, u2,t = dΔu2 + u1u2m is studied, where x ∈ R2, 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 |x| → ∞. 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. © World Scientific Publishing Company.

Publication Date

4-1-2006

Publication Title

Reviews in Mathematical Physics

Volume

18

Issue

3

Number of Pages

285-310

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1142/S0129055X06002681

Socpus ID

33745033386 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/33745033386

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