The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Joan Verdera - One of the best experts on this subject based on the ideXlab platform.
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the ellipse law kirchhoff meets dislocations
Communications in Mathematical Physics, 2020Co-Authors: Jose A Carrillo, Joan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Joan VerderaAbstract:In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \({\alpha \in \mathbb{R}}\). The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for \({\alpha \in (0, 1)}\) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes \({\sqrt{1-\alpha}}\) and \({\sqrt{1+\alpha}}\). This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result that domains enclosed by ellipses are Rotating Vortex patches, called Kirchhoff ellipses.
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the ellipse law kirchhoff meets dislocations
arXiv: Analysis of PDEs, 2017Co-Authors: Jose A Carrillo, Joan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Joan VerderaAbstract:In this paper we consider a nonlocal energy $I_\alpha$ whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter $\alpha\in \R$. The case $\alpha=0$ corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; $\alpha=1$ corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for $\alpha\in (0,1)$ the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an \emph{ellipse}. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are Rotating Vortex patches, called \emph{Kirchhoff ellipses}. Therefore we show a surprising connection between vortices and dislocations.
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boundary regularity of Rotating Vortex patches
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Taoufik Hmidi, Joan Mateu, Joan VerderaAbstract:We show that the boundary of a Rotating Vortex patch (or V-state, in the terminology of Deem and Zabusky) is C∞, provided the patch is close to the bifurcation circle in the Lipschitz norm. The Rotating patch is also convex if it is close to the bifurcation circle in the C2 norm. Our proof is based on Burbea’s approach to V-states.
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Boundary regularity of Rotating Vortex patches
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Taoufik Hmidi, Joan Mateu, Joan VerderaAbstract:We show that the boundary of a Rotating Vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The Rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on Hölder spaces, certain non-convolution singular integral operators of Calderón-Zygmund type.
Taoufik Hmidi - One of the best experts on this subject based on the ideXlab platform.
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existence of coRotating and counter Rotating Vortex pairs for active scalar equations
Communications in Mathematical Physics, 2017Co-Authors: Taoufik Hmidi, Joan MateuAbstract:In this paper, we study the existence of coRotating and counter-Rotating pairs of simply connected patches for Euler equations and the $${{\rm (SQG)}_{\alpha}}$$ equations with $${\alpha \in (0,1)}$$ . From the numerical experiments implemented for Euler equations in Deem and Zabusky (Phys Rev Lett 40(13):859–862, 1978), Pierrehumbert (J Fluid Mech 99:129–144, 1980), Saffman and Szeto (Phys Fluids 23(12):2339–2342, 1980) it is conjectured the existence of a curve of steady Vortex pairs passing through the point Vortex pairs. There are some analytical proofs based on variational principle (Keady in J Aust Math Soc Ser B 26:487–502, 1985; Turkington in Nonlinear Anal Theory Methods Appl 9(4):351–369, 1985); however, they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single Vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $${{\rm (SQG)}_{\alpha}}$$ equation when $${\alpha \in (0,1)}$$ . The proofs rely on the contour dynamics equations combined with a desingularization of the point Vortex pairs and the application of the implicit function theorem.
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existence of coRotating and counter Rotating Vortex pairs for active scalar equations
arXiv: Analysis of PDEs, 2016Co-Authors: Taoufik Hmidi, Joan MateuAbstract:In this paper, we study the existence of coRotating and counter-Rotating pairs of simply connected patches for Euler equations and the $(\hbox{SQG})_\alpha$ equations with $\alpha\in (0,1).$ From the numerical experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is conjectured the existence of a curve of steady Vortex pairs passing through the point Vortex pairs. There are some analytical proofs based on variational principle \cite{keady, Tur}, however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single Vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $(\hbox{SQG})_\alpha$ equation when $\alpha\in (0,1)$. The proofs rely on the contour dynamics equations combined with a desingularization of the point Vortex pairs and the application of the implicit function theorem.
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boundary regularity of Rotating Vortex patches
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Taoufik Hmidi, Joan Mateu, Joan VerderaAbstract:We show that the boundary of a Rotating Vortex patch (or V-state, in the terminology of Deem and Zabusky) is C∞, provided the patch is close to the bifurcation circle in the Lipschitz norm. The Rotating patch is also convex if it is close to the bifurcation circle in the C2 norm. Our proof is based on Burbea’s approach to V-states.
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Boundary regularity of Rotating Vortex patches
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Taoufik Hmidi, Joan Mateu, Joan VerderaAbstract:We show that the boundary of a Rotating Vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The Rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on Hölder spaces, certain non-convolution singular integral operators of Calderón-Zygmund type.
Joan Mateu - One of the best experts on this subject based on the ideXlab platform.
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the ellipse law kirchhoff meets dislocations
Communications in Mathematical Physics, 2020Co-Authors: Jose A Carrillo, Joan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Joan VerderaAbstract:In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \({\alpha \in \mathbb{R}}\). The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for \({\alpha \in (0, 1)}\) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes \({\sqrt{1-\alpha}}\) and \({\sqrt{1+\alpha}}\). This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result that domains enclosed by ellipses are Rotating Vortex patches, called Kirchhoff ellipses.
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the ellipse law kirchhoff meets dislocations
arXiv: Analysis of PDEs, 2017Co-Authors: Jose A Carrillo, Joan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Joan VerderaAbstract:In this paper we consider a nonlocal energy $I_\alpha$ whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter $\alpha\in \R$. The case $\alpha=0$ corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; $\alpha=1$ corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for $\alpha\in (0,1)$ the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an \emph{ellipse}. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are Rotating Vortex patches, called \emph{Kirchhoff ellipses}. Therefore we show a surprising connection between vortices and dislocations.
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existence of coRotating and counter Rotating Vortex pairs for active scalar equations
Communications in Mathematical Physics, 2017Co-Authors: Taoufik Hmidi, Joan MateuAbstract:In this paper, we study the existence of coRotating and counter-Rotating pairs of simply connected patches for Euler equations and the $${{\rm (SQG)}_{\alpha}}$$ equations with $${\alpha \in (0,1)}$$ . From the numerical experiments implemented for Euler equations in Deem and Zabusky (Phys Rev Lett 40(13):859–862, 1978), Pierrehumbert (J Fluid Mech 99:129–144, 1980), Saffman and Szeto (Phys Fluids 23(12):2339–2342, 1980) it is conjectured the existence of a curve of steady Vortex pairs passing through the point Vortex pairs. There are some analytical proofs based on variational principle (Keady in J Aust Math Soc Ser B 26:487–502, 1985; Turkington in Nonlinear Anal Theory Methods Appl 9(4):351–369, 1985); however, they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single Vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $${{\rm (SQG)}_{\alpha}}$$ equation when $${\alpha \in (0,1)}$$ . The proofs rely on the contour dynamics equations combined with a desingularization of the point Vortex pairs and the application of the implicit function theorem.
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existence of coRotating and counter Rotating Vortex pairs for active scalar equations
arXiv: Analysis of PDEs, 2016Co-Authors: Taoufik Hmidi, Joan MateuAbstract:In this paper, we study the existence of coRotating and counter-Rotating pairs of simply connected patches for Euler equations and the $(\hbox{SQG})_\alpha$ equations with $\alpha\in (0,1).$ From the numerical experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is conjectured the existence of a curve of steady Vortex pairs passing through the point Vortex pairs. There are some analytical proofs based on variational principle \cite{keady, Tur}, however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single Vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $(\hbox{SQG})_\alpha$ equation when $\alpha\in (0,1)$. The proofs rely on the contour dynamics equations combined with a desingularization of the point Vortex pairs and the application of the implicit function theorem.
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boundary regularity of Rotating Vortex patches
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Taoufik Hmidi, Joan Mateu, Joan VerderaAbstract:We show that the boundary of a Rotating Vortex patch (or V-state, in the terminology of Deem and Zabusky) is C∞, provided the patch is close to the bifurcation circle in the Lipschitz norm. The Rotating patch is also convex if it is close to the bifurcation circle in the C2 norm. Our proof is based on Burbea’s approach to V-states.
B Pouligny - One of the best experts on this subject based on the ideXlab platform.
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azimuthal instability of the radial thermocapillary flow around a hot bead trapped at the water air interface
Physics of Fluids, 2020Co-Authors: G Koleski, A Vilquin, J C Loudet, Thomas Bickel, B PoulignyAbstract:We investigate the radial thermocapillary flow driven by a laser-heated microbead in partial wetting at the water–air interface. Particular attention is paid to the evolution of the convective flow patterns surrounding the hot sphere as the latter is increasingly heated. The flow morphology is nearly axisymmetric at low laser power (P). Increasing P leads to symmetry breaking with the onset of counter-Rotating Vortex pairs. The boundary condition at the interface, close to no-slip in the low-P regime, turns about stress-free between the Vortex pairs in the high-P regime. These observations strongly support the view that surface-active impurities are inevitably adsorbed on the water surface where they form an elastic layer. The onset of Vortex pairs is the signature of a hydrodynamic instability in the layer response to the centrifugal forced flow. Interestingly, our study paves the way for the design of active colloids capable of achieving high-speed self-propulsion via Vortex pair generation at a liquid interface.
J C Loudet - One of the best experts on this subject based on the ideXlab platform.
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azimuthal instability of the radial thermocapillary flow around a hot bead trapped at the water air interface
Physics of Fluids, 2020Co-Authors: G Koleski, A Vilquin, J C Loudet, Thomas Bickel, B PoulignyAbstract:We investigate the radial thermocapillary flow driven by a laser-heated microbead in partial wetting at the water–air interface. Particular attention is paid to the evolution of the convective flow patterns surrounding the hot sphere as the latter is increasingly heated. The flow morphology is nearly axisymmetric at low laser power (P). Increasing P leads to symmetry breaking with the onset of counter-Rotating Vortex pairs. The boundary condition at the interface, close to no-slip in the low-P regime, turns about stress-free between the Vortex pairs in the high-P regime. These observations strongly support the view that surface-active impurities are inevitably adsorbed on the water surface where they form an elastic layer. The onset of Vortex pairs is the signature of a hydrodynamic instability in the layer response to the centrifugal forced flow. Interestingly, our study paves the way for the design of active colloids capable of achieving high-speed self-propulsion via Vortex pair generation at a liquid interface.
