The IVP for the evolution equation of wave fronts in chemical reactions in low-regularity Sobolev spaces
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Journal of Evolution Equations
The IVP for the evolution equation of wave fronts in chemical reactions in low-regularity Sobolev spaces Alysson Cunha
and Eduardo Alarcon
Abstract. In this work, we study the initial-value problem for an equation of evolution of wave fronts in chemical reactions. We show that the associated initial-value problem is locally and globally well-posed in Sobolev spaces H s (R), where s > 1/2. The well-posedness in critical space H˙ 1/2 (R) for small initial data is obtained. We also show that our result is sharp, in the sense that the flow-map data solution is not C 2 at origin, for s < 1/2. Furthermore, we study the behavior of the solutions when μ ↓ 0.
1. Introduction This paper is concerned with the initial-value problem (IVP), for the evolution equation of wave fronts in chemical reactions (WFCR) u t − ∂x2 u − μ(1 − ∂x2 )−1/2 u − 21 (∂x u)2 = 0, x ∈ R, t ≥ 0, (1.1) u(x, 0) = φ(x), where μ > 0 is a constant and u is a real-valued function. First, we present a derivation of Equation (1.1). Indeed, an initial-value problem equivalent to (1.1) ⎧ +∞ +∞ H (x , t) ⎪ 2 H − 1 (∂ H )2 − δ G ⎪ ∂ H − D ∂ ei k(x−x ) dx dk = 0, ⎪ C x ⎨ t 2 x 8π −∞ −∞ 1 + k2 ⎪ 4 ⎪ ⎪ ⎩u(x, 0) = φ(x), (1.2) where DC is dimensionless catalyst diffusivity, δ is relative density and G is dimensionless acceleration of gravity, was derived by Sivashinsky et al. [32], to describe vertical propagation of chemical wave fronts in the presence of instability due to density gradients (possibly thermally induced). Assuming an interaction region thin enough to be described as a surface (z = H (x, y, t)), where H is the vertical position of the front, they use thermo-hydrodynamic equations in the regions of reacted fluid (z < H (x, y, t)) and unreacted fluid ( z > H (x, y, t)) together with conservation of energy, matter and momentum to derive jump conditions on discontinuities
A. Cunha and E. Alarcon
J. Evol. Equ.
at the interface. The equations governing these autocatalytic systems involving propagating reaction–diffusion fronts have been derived in [18], where they consider the reaction front to be very thin chemically, and other assumption in use involves how the densities of the fluids change with temperature. Since the density changes due to thermal expansion of the fluids are small, write the density of the fluids to first order as ρ(T ) = ρ1 [1 − α(T − T1 )], where ρ(T ) is the density at temperature T , ρ1 is density at the reference temperature T1 and α is the classical thermal expansion coefficient at constant pressure. The relative difference between the densities of these two fluids at ρa − ρb , the front is one of the key parameters in this study and is defined by δ = ρb a b where ρ and ρ are the densities of the fluid above the front (unreacted fluid) and of that below the front (reacted fluid), respectively. This is due to the fact that ρ is dependent on the thermal diffusivity of the fluids, and in [32], the diffusivity is assumed to be infinite. As in [18], they obtain the following system of
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