In this paper, we consider the three-dimensional Cauchy problem for the viscous electron magnetohydrodynamics (EMHD) system. We show that if a very weak solution b satisfies <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nabla \times b\in L^{4}(0,T;L^{\frac{12}{5}}(\mathbb {R}^{3}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mi>b</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>4</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>L</mi> <mfrac> <mn>12</mn> <mn>5</mn> </mfrac> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then it automatically belongs to the natural energy space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{\infty }(0,T;L^{2})\cap L^{2}(0,T;H^{1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and obeys the energy equality on [0, T]. The proof relies on the Biot-Savart law and the refined nonlinear estimates.

On the energy equality for very weak solutions to the EMHD system

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发表日期：

2026年6月23日

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Fan Wu

摘要

In this paper, we consider the three-dimensional Cauchy problem for the viscous electron magnetohydrodynamics (EMHD) system. We show that if a very weak solution b satisfies $\nabla \times b\in L^{4}(0,T;L^{\frac{12}{5}}(\mathbb {R}^{3}))$ $\nabla \times b \in L^{4} (0, T; L^{\frac{12}{5}} (R^{3}))$ , then it automatically belongs to the natural energy space $L^{\infty }(0,T;L^{2})\cap L^{2}(0,T;H^{1})$ $L^{\infty} (0, T; L^{2}) \cap L^{2} (0, T; H^{1})$ and obeys the energy equality on [0, T]. The proof relies on the Biot-Savart law and the refined nonlinear estimates.