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Register a new userRate of convergence to self-similarity for Smoluchowskis coagulation equation with constant coefficients
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We show that solutions to Smoluchowskis equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the L^2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.
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Existence and non-existence of integrable stationary solutions to Smoluchowskis coagulation equation with source are investigated when the source term is integrable with an arbitrary support in (0, $infty$). Besides algebraic upper and lower bounds, a monotonicity condition is required for the coagulation kernel. Connections between integrability properties of the source and the corresponding stationary solutions are also studied.
We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d 2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability $alpha$ $in$ (0, 1) or collide elastically with probability 1 -- $alpha$. Such equation is highly dissipative in the sense that all observables, hence solutions, vanish as time progresses. Following a contribution , by two of the authors, considering well posedness of the steady self-similar profile in the regime of small annihilation rate $alpha$ $ll$ 1, we prove here that such self-similar profile is the intermediate asymptotic attractor to the annihilation dynamics with explicit universal algebraic rate. This settles the issue about universality of the annihilation rate for this model brought in the applied literature.
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