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SUMMARY:The some solution of the Bryan-Pidduck equation
DTSTART;VALUE=DATE-TIME:20260922T143000Z
DTEND;VALUE=DATE-TIME:20260922T143500Z
DTSTAMP;VALUE=DATE-TIME:20260914T013546Z
UID:indico-contribution-513@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Oleksii Hukalov (B.Verkin Institute for Low Temperat
 ure Physics and Engineering of the NAS of Ukraine)\nThe Boltzmann equation
  [1] that describes the evolution of rarefied gases is one of the main equ
 ations of the kinetic theory of gases. For a model of rough spheres\, the 
 equation has the form:\n\\begin{equation}\nD(f)= Q(f\,f)\,\n\\end{equation
 }\nwhere the left-hand side of the equation is the  differential operator:
 \n$$ D(f)\\equiv\\frac{\\partial f}{\\partial t}+\\left(V\,\\frac{\\partia
 l f}{\\partial x}\\right)\,$$\nand the right-hand side of the Boltamann eq
 uation  is the collision integral\, which for the hard spheres model is as
  follows:\n$$Q(f\,f)\\equiv\\frac{d^{2}}{2}\\int_{R^{3}}\\!\\!\\!dV_{1} \\
 int_{R^{3}}\\!\\!\\!d\\omega_{1} \\int_{\\Sigma} d\\alpha B(V-V_{1}\,\\alp
 ha)\\Bigl[f(t\,V_{1}^{*}\,x\,\\omega_{1}^{*})f(t\,V^{*}\,x\,\\omega^{*})-f
 (t\,V\,x\,\\omega)f(t\,V_{1}\,x\,\\omega_{1})\\Bigl]\,$$ where $f(t\,V\,x\
 ,\\omega)$ is the distribution function of particles.\n\nThe problem of de
 termination of the exact and approximate solutions of the Boltzmann equati
 on in the explicit form is quite urgent. At present\, the sole known exact
  solution of the Boltzmann equation is an expression usually called the Ma
 xwell distribution or simply Maxwellian (after J. C. Maxwell\, Scottish ph
 ysicist). In the case of Maxwellians $M$\, we get\n\\begin{equation}\nD(f)
 =0\, \\quad  Q(f\,f)=0.\n\\end{equation}\n\nThe solution to this equation 
 will be look for in the next form:\n$$ f(t\,x\,V\,\\omega)=\\sum\\limits_{
 i=1}^{\\infty}\\varphi_i(t\,x)M_i(t\,x\,V\,\\omega).$$\n\nAs a measure of 
 the deviation between the parts of the Boltzmann equation we will consider
  a uniform-integral error of the form:\n$$\\Delta=\\sup_{(t\,x)\\in \\math
 bb{R}^4}\\int_{\\mathbb{R}^3}dV \\int_{\\mathbb{R}^3}d\\omega\\Big|D(f)-Q(
 f\,f)\\Big|.\n$$\n\nIn the paper[2]\, we were obtained sufficient conditio
 ns for the coefficient functions and hydrodynamic parameters appearing in 
 the distribution\, which enable one to make the analyzed error as small as
  desired.\n\n**REFERENCES**\n\n[1] S. Chapman and T.G. Cowling. The Mathem
 atical Theory of Non-Uniform Gases. *Cambridge Univ. Press\,* Cambridge\, 
 1952.\n\n[2] V. Gordevskyy and O. Hukalov\; The Interaction of a Countable
  Number of Eddy Flows for the Bryan-Pidduck Model. *J. Math. Phys. Anal. G
 eom.* 2025\, 21\, 218-231.\n\nhttps://indico.bitp.kiev.ua/event/18/contrib
 utions/513/
LOCATION:Bogolyubov Institute for Theoretical Physics Conference Hall
URL:https://indico.bitp.kiev.ua/event/18/contributions/513/
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