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Register a new userLocal well-posedness for Boltzmanns equation and the Boltzmann hierarchy via Wigner transform
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We use the dispersive properties of the linear Schr{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $mathbb{R}^d$ for $dgeq 2$. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data $f_0$ are weight
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In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $mathbb{R}^d$, $dgeq 2$, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $left< v right>^beta f_0 in L^2_v H^alpha_x$ for $min (alpha,beta) > frac{d-1}{2}$. We prove that if $alpha,beta$ are large enough, then it is possible to propagate moments in $x$ and derivatives in $v$ (for instance, $left< x right>^k left< abla_v right>^ell f in L^infty_T L^2_{x,v}$ if $f_0$ is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of $f$. We also prove a persistence of regularity result for the scale of Sobolev spaces $H^{alpha,beta}$; and, continuity of the solution map in $H^{alpha,beta}$. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the $H$-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in $H^{alpha,beta}$ for any $alpha,beta > frac{d-1}{2}$, without any additional regularity or decay assumptions.
We provide a new analysis of the Boltzmann equation with constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove global well-posedness and a version of scattering, assuming that the data $f_0$ is sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision gain term in Boltzmanns equation, combined with a novel application of the Kaniel-Shinbrot iteration.
In this paper, we address the local well-posedness of the spatially inhomogeneous non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $gamma + 2s < 0$. Our main result completes the picture for local well-posedness in this decay class by removing the restriction $gamma + 2s > -3/2$ of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when $gamma in (-3,0]$ and $s in (0,1/2)$ in a weighted $C^1$ space that we include as well.
The Vlasov-Poisson-Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We consider the system in a convex domain with the Cercignani-Lampis boundary condition. We construct a uniqueness local-in-time solution based on an $L^infty$-estimate and $W^{1,p}$-estimate. In particular, we develop a new iteration scheme along the characteristic with the Cercignani-Lampis boundary for the $L^infty$-estimate, and an intrinsic decomposition of boundary integral for $W^{1,p}$-estimate.
In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig cite{IK} and cite{IK2} to approach the problem in a perturbative way.
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