<p>The paper is devoted to analytical and numerical study of solutions to the Vlasov–Poisson–Landau kinetic equations (VPLE) for distribution functions with typical length <i>L</i> such that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \varepsilon = r_D/L \ll 1,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>=</mo> <msub> <mi>r</mi> <mi>D</mi> </msub> <mo stretchy="false">/</mo> <mi>L</mi> <mo>≪</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( r_D \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation> stands for the Debye radius. It is also assumed that the Knudsen number is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( \mathrm{K\!n} = l/L = O(1),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="normal">K</mi> <mspace width="-0.166667em" /> <mi mathvariant="normal">n</mi> </mrow> <mo>=</mo> <mi>l</mi> <mo stretchy="false">/</mo> <mi>L</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <i>l</i> is the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( \varepsilon &gt; 0.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>&gt;</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is known that the formal limit of VPLE for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( \varepsilon = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> does not describe a rapidly oscillating part of the electric field. Our aim is to study the behavior of the “true” electric field near this limit. We consider the problem with standard Maxwellian initial conditions isotropic in velocities and show that there is almost no damping of these oscillations in the collisionless case. We derive approximate formula for the electric field and then confirm it numerically by using a simplified Bathnagar–Gross–Krook (BGK-type) model of Vlasov–Poisson–Landau equations (VPLE). Also we consider another class of initial conditions that leads to strong oscillations having the amplitude of order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( O(1/\varepsilon ) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We study numerical solutions of that class for different values of parameters <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( \varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\( \mathrm{K\!n}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="normal">K</mi> <mspace width="-0.166667em" /> <mi mathvariant="normal">n</mi> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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ASYMPTOTIC SOLUTIONS OF THE VLASOV–POISSON–LANDAU KINETIC EQUATIONS

  • A. V. Bobylev,
  • I. F. Potapenko

摘要

The paper is devoted to analytical and numerical study of solutions to the Vlasov–Poisson–Landau kinetic equations (VPLE) for distribution functions with typical length L such that \( \varepsilon = r_D/L \ll 1,\) ε = r D / L 1 , where \( r_D \) r D stands for the Debye radius. It is also assumed that the Knudsen number is \( \mathrm{K\!n} = l/L = O(1),\) K n = l / L = O ( 1 ) , where l is the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small \( \varepsilon > 0.\) ε > 0 . It is known that the formal limit of VPLE for \( \varepsilon = 0\) ε = 0 does not describe a rapidly oscillating part of the electric field. Our aim is to study the behavior of the “true” electric field near this limit. We consider the problem with standard Maxwellian initial conditions isotropic in velocities and show that there is almost no damping of these oscillations in the collisionless case. We derive approximate formula for the electric field and then confirm it numerically by using a simplified Bathnagar–Gross–Krook (BGK-type) model of Vlasov–Poisson–Landau equations (VPLE). Also we consider another class of initial conditions that leads to strong oscillations having the amplitude of order \( O(1/\varepsilon ) \) O ( 1 / ε ) . We study numerical solutions of that class for different values of parameters \( \varepsilon \) ε and \( \mathrm{K\!n}.\) K n .