Local Well-Posedness and Sensitivity Analysis for the Self-Organized Kinetic Model

Jiang, Ning; Zeng, Zhang

doi:10.1007/s10440-021-00457-8

Cited by 1 publication

(2 citation statements)

References 27 publications

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“…Starting with the model (1.1) (actually in the second case, see below), they proposed the continuum model through considering some parameter tending to zero [1]. Moreover, describing the agents by some statistical distribution of their speed and positions is also referred as the kinetic model or mean-field model, and some results of well-posedness in this model have been obtained [19][20][21][22][23][24].…”

Section: Brief Reviewmentioning

confidence: 99%

“…Since

{u}_2\left[f\right]&amp;#x0003D;\frac{J\left[f\right]}{\sqrt{{\left&amp;#x0007C;J\left[f\right]\right&amp;#x0007C;}&amp;#x0005E;2}},\left(J\left[f\right]&amp;#x0003D;{\int}_{{\mathrm{\mathbb{R}}}&amp;#x0005E;3} fv\mathrm{d}v\right)

, we can calculate that (see [22])

\begin{matrix} alignleft \\ align-1 & align-2 \partial_{x} u_{2} [f] = \frac{\partial_{x} J [f]}{\sqrt{{|J [f]|}^{2}}} - \frac{J [f] J [f] \partial_{x} J [f]}{{(\sqrt{{|J [f]|}^{2}})}^{3}}, \\ align-1 & align-2 \partial_{x}^{s} u_{2} [f] = \frac{\partial_{x}^{s} J [f]}{\sqrt{{|J [f]|}^{2}}} + \frac{\partial_{x}^{s} J [f] P_{2}^{s} (J [f])}{{(\sqrt{{|J [f]|}^{2}})}^{3}} + \end{matrix}

…”

Section: A Priori Estimates and Local Existence For Case (Ii)mentioning

confidence: 99%

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Local well‐posedness of classical solutions for the kinetic self‐organized model of Cucker–Smale type

Zhou,

Zhang,

2024

Math Methods in App Sciences

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We study in this paper the local well‐posedness of classical solutions to the Cauchy problem for a kinetic self‐organized model of Cucker–Smale type for collective motions, which includes two cases of velocity alignment mechanisms: normalized and nonnormalized reorientation cases. The main concern is the a priori estimates for both cases. Our treatments rely on two key ingredients: One point is to deal with the possible singularity and high nonlinearity arising from the normalized/nonnormalized mean velocity, and the other is to control the growth with respect to the microscopic velocity variable in the whole space by employing weighted estimates.

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