Scholarlink

<p>We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma \in [-3,-2]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mn>3</mn> <mo>,</mo> <mo>-</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that solutions can be continued as long as the following three quantities remain finite, uniformly in <i>t</i> and <i>x</i>: (1) the mass density, (2) the velocity moment of order <i>s</i> for any small <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(s&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and (3) the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^p_v\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>v</mi> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> norm for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(p&gt;3/(5+\gamma )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, we do not require a bound on the energy density.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

A Continuation Criterion for the Landau Equation with Very Soft and Coulomb Potentials

  • 期刊论文

  • 发表日期：

    2026年3月25日

查看全文
  • Stanley Snelson,
  • Caleb Solomon

摘要

We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, \(\gamma \in [-3,-2]\) γ ∈ [ - 3 , - 2 ] . We show that solutions can be continued as long as the following three quantities remain finite, uniformly in t and x: (1) the mass density, (2) the velocity moment of order s for any small \(s>0\) s > 0 , and (3) the \(L^p_v\) L v p norm for any \(p>3/(5+\gamma )\) p > 3 / ( 5 + γ ) . In particular, we do not require a bound on the energy density.