<p>We study the initial value problem for a system of equations describing the motion of two-dimensional non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. We consider the complete odd viscous stress tensor with a general density-dependent viscosity coefficient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(\rho )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Under suitable assumptions, we prove the local existence and uniqueness of strong solutions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${H^s}(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(s>2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>></mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, for a class of viscosity coefficients covering the particular case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(\rho )=a\rho ^\alpha +b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <msup> <mi>ρ</mi> <mi>α</mi> </msup> <mo>+</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(a,b,\alpha )\in \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, generalising the result of [26] devoted to the case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f(\rho )=\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we are able to do so without requiring the initial density variation to belong to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${L^2}(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. As a major step of the proof, we exhibit an effective velocity for this sytem, generalising the so-called “Elsässer formulation” recently derived in [27].</p>

Well-posedness for 2D Non-homogeneous Incompressible Fluids with General Density-Dependent Odd Viscosity

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发表日期：

2026年6月18日

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Matthieu Pageard

摘要

We study the initial value problem for a system of equations describing the motion of two-dimensional non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. We consider the complete odd viscous stress tensor with a general density-dependent viscosity coefficient $f(\rho )$ $f (ρ)$ . Under suitable assumptions, we prove the local existence and uniqueness of strong solutions in ${H^s}(\mathbb {R}^2)$ $H^{s} (R^{2})$ $(s>2)$ $(s > 2)$ , for a class of viscosity coefficients covering the particular case $f(\rho )=a\rho ^\alpha +b$ $f (ρ) = a ρ^{α} + b$ for any $(a,b,\alpha )\in \mathbb {R}^3$ $(a, b, α) \in R^{3}$ , generalising the result of [26] devoted to the case $f(\rho )=\rho $ $f (ρ) = ρ$ . Additionally, we are able to do so without requiring the initial density variation to belong to ${L^2}(\mathbb {R}^2)$ $L^{2} (R^{2})$ . As a major step of the proof, we exhibit an effective velocity for this sytem, generalising the so-called “Elsässer formulation” recently derived in [27].