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• W4386475115 abstract "We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper M comma g right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(M,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis u Subscript epsilon comma delta Baseline minus delta normal upper Delta u Subscript epsilon comma delta Baseline right-parenthesis plus d i v German f Subscript epsilon Baseline left-parenthesis bold x comma u Subscript epsilon comma delta Baseline right-parenthesis d t equals epsilon normal upper Delta u Subscript epsilon comma delta Baseline d t plus normal upper Phi left-parenthesis bold x comma u Subscript epsilon comma delta Baseline right-parenthesis d upper W Subscript t Baseline comma> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>f</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} d left (u_{varepsilon ,delta } -delta Delta u_{varepsilon ,delta }right ) +divmathfrak {f}_{varepsilon }(mathbf {x}, u_{varepsilon ,delta }), dt =varepsilon Delta u_{varepsilon ,delta }, dt + Phi (mathbf {x}, u_{varepsilon ,delta }), dW_t, end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German f Subscript epsilon> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>f</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ε<!-- ε --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mathfrak {f}_{varepsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sequence of smooth vector fields converging in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript p Baseline left-parenthesis upper M times double-struck upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^p(Mtimes mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 2> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon down-arrow 0> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo stretchy=false>↓<!-- ↓ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon downarrow 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> towards a vector field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German f element-of upper L Superscript p Baseline left-parenthesis upper M semicolon upper C Superscript 1 Baseline left-parenthesis double-struck upper R right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>f</mml:mi> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {f}in L^p(M;C^1(mathbb {R}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W Subscript t> <mml:semantics> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>W_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Wiener process defined on a filtered probability space. First, for fixed values of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=application/x-tex>varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we establish the existence and uniqueness of weak solutions to the Cauchy problem for the above-stated equation. Assuming that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German f> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>f</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is non-degenerate and that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=application/x-tex>varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tend to zero with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta slash epsilon squared> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>ε<!-- ε --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>delta /varepsilon ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded, we show that there exists a subsequence of solutions that strongly converges in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript omega comma t comma bold x Superscript 1> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>L^1_{omega ,t,mathbf {x}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a martingale solution of the following stochastic conservation law with discontinuous flux: <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d u plus d i v German f left-parenthesis bold x comma u right-parenthesis d t equals normal upper Phi left-parenthesis u right-parenthesis d upper W Subscript t Baseline period> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>f</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} d u +divmathfrak {f}(mathbf {x}, u),dt =Phi (u), dW_t. end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> The proofs make use of Galerkin approximations, kinetic formulations as well as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-measures and new velocity averaging results for stochastic continuity equations. The analysis relies on the use of a.s. representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws." @default.
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• W4386475115 title "A dynamic capillarity equation with stochastic forcing on manifolds: A singular limit problem" @default.
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