Radon measure-valued solutions of first order scalar conservation laws
Radon measure-valued solutions of first order scalar conservation laws
Abstract We study nonnegative solutions of the Cauchy problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>{</m:mo><m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"><m:mtr><m:mtd /><m:mtd columnalign="left"><m:mrow><m:mrow><m:mrow><m:msub><m:mo>∂</m:mo><m:mi>t</m:mi></m:msub><m:mo></m:mo><m:mi>u</m:mi></m:mrow><m:mo>+</m:mo><m:mrow><m:msub><m:mo>∂</m:mo><m:mi>x</m:mi></m:msub><m:mo></m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mi>φ</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>u</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo stretchy="false">]</m:mo></m:mrow></m:mrow></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow></m:mtd><m:mtd /><m:mtd columnalign="right"><m:mrow><m:mrow><m:mrow><m:mtext>in </m:mtext><m:mo></m:mo><m:mi>ℝ</m:mi></m:mrow><m:mo>×</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>T</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow></m:mtd></m:mtr><m:mtr><m:mtd /><m:mtd columnalign="left"><m:mrow><m:mi>u</m:mi><m:mo>=</m:mo><m:msub><m:mi>u</m:mi><m:mn>0</m:mn></m:msub><m:mo>≥</m:mo><m:mn>0</m:mn></m:mrow></m:mtd><m:mtd /><m:mtd columnalign="right"><m:mrow><m:mrow><m:mrow><m:mtext>in </m:mtext><m:mo></m:mo><m:mi>ℝ</m:mi></m:mrow><m:mo>×</m:mo><m:mrow><m:mo stretchy="false">{</m:mo><m:mn>0</m:mn><m:mo stretchy="false">}</m:mo></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow></m:mtd></m:mtr></m:mtable></m:mrow></m:math> \left\{\begin{aligned} &\displaystyle\partial_{t}u+\partial_{x}[\varphi(u)]=0&% &\displaystyle\phantom{}\text{in }\mathbb{R}\times(0,T),\\ &\displaystyle u=u_{0}\geq 0&&\displaystyle\phantom{}\text{in }\mathbb{R}% \times\{0\},\end{aligned}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>u</m:mi><m:mn>0</m:mn></m:msub></m:math> {u_{0}} is a Radon measure and …