Lp-stability of solutions to Boltzmann equation with external forcing

帶外力的 Boltzmann 方程解的 Lp 穩定性

Student thesis: Master's Thesis

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  • Chi Honn CHENG

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Award date2 Oct 2009


The study of Cauchy problem of the Boltzmann equation is important in both theory and applications. When there is no external force, existence of global solutions and uniform stability of solutions in Lp space, where 1 ≤ p < ∞, were introduced in previous works of Boltzmann equation. In this thesis, we will investigate the uniform stability of solutions for the Cauchy problem of the Boltzmann equation when there is an external force in the case of soft potentials for 1 ≤ p < ∞. In the first part, we deal with the proof of uniform stability of solutions when the external force may be large in L1 space for soft potentials. L1 stability in the case of small external force was introduced in previous works of Boltzmann equation. Here, we establish the uniform L1 stability by deriving a Gronwall type estimates for L1 distance using the dispersion estimates of solutions to Boltzamnn equation with soft potentials near vacuum. In the second part, we discuss about the Lp stability of solutions for soft potentials when p > 1 under the influence of two kinds of external forces whose size can be small or large. When the external force is small, we first consider the time derivative of the p-th power of Lp norm of distance between two mild solutions f and g, then define nonlinear functionals, and apply the Gronwall's inequality to obtain the Lp stability. For the case of large external force, we can use the same method as in the case of L1. In other words, we use the constructive assumptions on external force and Gronwall's inequality to derive the Lp stability estimate. In addition, Minkowski's inequality is also used in this case.

    Research areas

  • Cauchy problem, Mathematical models, Kinetic theory of gases, Transport theory