Manuel del Pino

We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant $g>0$ g > 0 is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, Hélein, and Pacard (Pac. J. Math. 250:319–334, 2011). The method developed here is related to the construction of constant mean curvature surfaces through gluing, and can be applied to other overdetermined elliptic problems.

We consider the 2-dimensional harmonic map flow into $ \mathbb{S}^2 $ $ \begin{align} \left\{\begin{array}{ll} u_t = \Delta u + |\nabla u|^2u\quad\text{ in }\Omega\times (0, \infty), \\ u = \varphi \quad\text{ on }\partial\Omega\times (0, \infty), \\ u(\cdot, 0) = u_0\quad\text{ in }\Omega \end{array} \right. \end{align}~~~~(1) $ for a function $ u:\Omega\times[0, \infty)\to \mathbb{S}^2 $, where $ \Omega $ is a bounded domain in $ \mathbb{R}^2 $ with smooth boundary and $ u_0:\overline\Omega \to \mathbb{S}^2 $ is smooth satisfying $ \varphi = u_0\big|_{\partial\Omega} $. Let $ q_1 $, $ \cdots $, $ q_k $ be distinct points in $ \Omega $. We prove the existence of an initial datum $ u_0 $, $ u_*\in H^1(\Omega)\cap C(\overline{\Omega}) $, and smooth functions $ \xi_j(t)\to q_j $, $ \alpha_j(t)\to 0 $, $ 0<\mu_j(t)\to 0 $ (as $ t\to +\infty $), $ j = 1, \cdots, k $, such that the solution $ u_q $ of (1) has the form $ \begin{align} u_q(x, t) \approx u_*(x) + \sum\limits_{j = 1}^k Q_{\alpha_j(t)}\left(\omega\left(\frac{x-\xi_j(t)}{\mu_j(t)}\right)- \omega_\infty\right), \end{align} $ where the profile $ \omega = \mathcal{S}(x^2), x\in \mathbb{R}^2 $ is the 2-corrotation harmonic map, $ Q_{\alpha_j(t)} $ is the $ \alpha_j(t) $-rotation matrix around the third axis, and $ \omega_\infty = (0, 0, 1)^\top $. Furthermore, $ u_*(x) $ is a harmonic map from $ \Omega $ into $ \mathbb{S}^2 $ and the dilation parameters $ \mu_j(t) $ decay to $ 0 $ exponentially for $ j = 1, \cdots, k $.

The liquid drop model was introduced by Gamow in 1928 and Bohr–Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface $$\Sigma =\partial \Omega $$ Σ = ∂ Ω in $${\mathbb {R}}^3$$ R 3 that is critical for the energy $$\begin{aligned} {\mathcal {E}} (\Omega ) = {{{\textrm{Per}}}\,} (\Omega ) + \frac{1}{2} \int _\Omega \int _\Omega \frac{{\text {d}}x{\text {d}}y}{|x-y|} \end{aligned}$$ E ( Ω ) = Per ( Ω ) + 1 2 ∫ Ω ∫ Ω d x d y | x - y | under the volume constraint $$|\Omega | = m$$ | Ω | = m . The term $$\mathrm{Per\,} (\Omega ) $$ Per ( Ω ) corresponds to the surface area of $$\Sigma $$ Σ . The associated Euler–Lagrange equation is $$\begin{aligned} H_\Sigma (x) + \int _{\Omega } \frac{{\text {d}}y}{|x-y|} = \lambda \quad \hbox { for all } x\in \Sigma , \quad \end{aligned}$$ H Σ ( x ) + ∫ Ω d y | x - y | = λ for all x ∈ Σ , where $$H_\Sigma $$ H Σ stands for the mean curvature of the surface, and where $$\lambda \in {\mathbb {R}}$$ λ ∈ R is the Lagrange multiplier associated to the constraint $$|\Omega |=m$$ | Ω | = m . Round spheres enclosing balls of volume m are always solutions; they are minimizers for sufficiently small m . Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a “pearl necklace” with an axis located on a large circle, with a shape close to a Delaunay’s unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since, for the closely related constant mean curvature problem $$H_\Sigma = \lambda $$ H Σ = λ , the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $ω_{\varepsilon}(x,t)$ concentrated around points $ξ_{j}(t)$ that converge to a sum of Dirac delta masses as $\varepsilon\to0$. These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points $ξ_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $ξ_{j}$.

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W(x)=w(r)e^{iθ}$. Using explicit representation formulae for the Fourier modes in $θ$, we obtain sharp estimates for the inverse of the linearized operator which hold for a large class of right-hand sides. This theory can be applied, for example, to estimate the inverse after dropping the usual orthogonality conditions.

The liquid drop model was introduced by Gamow in 1928 and Bohr-Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface $Σ=\partial Ω$ in $\mathbb{R}^3$ that is critical for the energy $$ E(Ω) = {\rm Per\,} (Ω) + \frac 12 \int_Ω\int_Ω\frac {dxdy}{|x-y|} $$ under the volume constraint $|Ω| = m$. The term ${\rm Per\,} (Ω) $ corresponds to the surface area of $Σ$. The associated Euler-Lagrange equation is $$ H_Σ(x) + \int_{Ω} \frac {dy}{|x-y|} = λ\quad \hbox{ for all } x\in Σ, \quad $$ where $H_Σ$ stands for the mean curvature of the surface, and where $λ\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|Ω|=m$. Round spheres enclosing balls of volume $m$ are always solutions. They are minimizers for sufficiently small $m$. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a "pearl necklace" with an axis located on a large circle, with a shape close to a Delaunay's unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since for the closely related constant mean curvature problem $H_Σ= λ$, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant $g>0$ is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, Hélein, and Pacard \cite{hauswirth-helein-pacard}. The method developed here is related to the construction of constant mean curvature surfaces through gluing.