<p>This work aims to study the magnetohydrodynamic (MHD) equations of a two-dimensional barotropic, compressible flow in a bounded domain with a compact Lipschitz boundary. More specifically, the finite energy weak solutions of the proposed problem will be considered on a certain interval of time, and the global existence will be studied using essential tools from Orlicz space theory. The pressure acting on the fluid satisfying the assumptions that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_953_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">$p(\rho )=A\rho \log ^{\aleph }(\rho )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mi>ρ</mi> <msup> <mo>log</mo> <mi>ℵ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_953_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\aleph > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℵ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_953_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$A > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and large <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_953_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>.</p>

On the Existence of Two-Dimensional Solutions to the Magnetohydrodynamic Equations of a Compressible Flow in Orlicz Spaces

期刊论文
发表日期：

2025年3月3日

查看全文

Jan Muhammad

摘要

This work aims to study the magnetohydrodynamic (MHD) equations of a two-dimensional barotropic, compressible flow in a bounded domain with a compact Lipschitz boundary. More specifically, the finite energy weak solutions of the proposed problem will be considered on a certain interval of time, and the global existence will be studied using essential tools from Orlicz space theory. The pressure acting on the fluid satisfying the assumptions that $p(\rho )=A\rho \log ^{\aleph }(\rho )$ $p (ρ) = A ρ {log}^{ℵ} (ρ)$ with $\aleph > 1$ $ℵ > 1$ , $A > 0$ $A > 0$ and large $\rho $ $ρ$ .