<p>For mobility exponents <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \in (2,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we prove the existence of strictly positive solutions to stochastic thin-film equations with singular effective interface potential and Stratonovich-type lower-order terms. With the perspective of using these solutions in Part II to construct surface-tension energy-dissipating solutions to stochastic thin-film equations with compactly supported initial data, for which finite speed of propagation is shown in Grün and Klein (SIAM J Math Anal), we establish decay estimates on the sum of surface-tension energy and effective interface potential—without relying on further functionals involving initial data. Besides an extension of earlier techniques used in the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and a refinement of oscillation estimates for discrete solutions, the main analytical novelty of this paper is a discretization method, which shows nonnegativity for a finite-element counterpart of the integral <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\int _{{\mathcal {O}}}}(u^{n-2}u_{x})_x u_{xx} \textrm{d}x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∫</mo> <mi mathvariant="script">O</mi> </msub> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </msub> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mtext>d</mtext> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> under periodic boundary conditions in the parameter regime <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n \in (2,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This nonnegativity property serves to control Itô-correction terms in the estimate for the decay of the surface-tension energy. This way, it is the key to obtain the desired decay estimates for the sum of surface-tension energy and effective interface potential, which permit to establish the singular limit of vanishing effective interface potential in Part II.</p>

Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: part I—positive solutions

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2025年9月18日

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Günther Grün,
Lorenz Klein

摘要

For mobility exponents $n \in (2,3)$ $n \in (2, 3)$ , we prove the existence of strictly positive solutions to stochastic thin-film equations with singular effective interface potential and Stratonovich-type lower-order terms. With the perspective of using these solutions in Part II to construct surface-tension energy-dissipating solutions to stochastic thin-film equations with compactly supported initial data, for which finite speed of propagation is shown in Grün and Klein (SIAM J Math Anal), we establish decay estimates on the sum of surface-tension energy and effective interface potential—without relying on further functionals involving initial data. Besides an extension of earlier techniques used in the case $n=2$ $n = 2$ and a refinement of oscillation estimates for discrete solutions, the main analytical novelty of this paper is a discretization method, which shows nonnegativity for a finite-element counterpart of the integral ${\int _{{\mathcal {O}}}}(u^{n-2}u_{x})_x u_{xx} \textrm{d}x$ $\int_{O} {(u^{n - 2} u_{x})}_{x} u_{xx} d x$ under periodic boundary conditions in the parameter regime $n \in (2,3)$ $n \in (2, 3)$ . This nonnegativity property serves to control Itô-correction terms in the estimate for the decay of the surface-tension energy. This way, it is the key to obtain the desired decay estimates for the sum of surface-tension energy and effective interface potential, which permit to establish the singular limit of vanishing effective interface potential in Part II.