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This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $\omega_{\varepsilon}(x,t)$ concentrated around points $\xi_{j}(t)$ 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 $\xi_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $\xi_{j}$.

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W(x)=w(r)e^{i\theta}$. Using explicit representation formulae for the Fourier modes in $\theta$, 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., Comment: 21 pages

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 $\Sigma =\partial \Omega$ in $\mathbb{R}^3$ that is critical for the energy $$ E(\Omega) = {\rm Per\,} (\Omega ) + \frac 12 \int_\Omega\int_\Omega \frac {dxdy}{|x-y|} $$ under the volume constraint $|\Omega| = m$. The term ${\rm Per\,} (\Omega ) $ corresponds to the surface area of $\Sigma$. The associated Euler-Lagrange equation is $$ H_\Sigma (x) + \int_{\Omega } \frac {dy}{|x-y|} = \lambda \quad \hbox{ for all } x\in \Sigma, \quad $$ where $H_\Sigma$ stands for the mean curvature of the surface, and where $\lambda\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|\Omega|=m$. 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 $H_\Sigma = \lambda$, 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, H\'elein, and Pacard \cite{hauswirth-helein-pacard}. The method developed here is related to the construction of constant mean curvature surfaces through gluing., Comment: 93 pages

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-\Delta_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2\pi} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8\pi

A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $\omega(x,t) = W(x_1-ct,x_2 )$ where $W(x)$ is compactly supported and odd in $x_2$. We revisit the problem of constructing solutions which are highly $\varepsilon$-concentrated around points $ (0, \pm q)$, more precisely with approximately radially symmetric, compactly supported bumps with radius $\varepsilon$ and masses $\pm m$. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters $q$ and $\varepsilon$ for the solution and its propagation speed $c$ are established. These results improve constructions through variational methods in [14] and in [5] for the case of a bounded domain., Comment: To appear in Journal of Differential Equations. arXiv admin note: substantial text overlap with arXiv:2310.07238

We consider the problem of finding a solution to the incompressible Euler equations $$ \omega_t + v\cdot \nabla \omega = 0 \quad \hbox{ in } \mathbb{R}^2 \times (0,\infty), \quad v(x,t) = \frac 1{2\pi} \int_{{\mathbb R}^2} \frac {(y-x)^\perp}{|y-x|^2} \omega (y,t)\, dy $$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$ \omega (x,t) = \omega_0(x-ct\, e ) - \omega_0 ( x+ ct \, e) + o(1) \ \hbox{ as } t\to\infty $$ where $$ \omega_0( x ) = \frac 1{\varepsilon^{2}} W \left ( \frac {x-q} \varepsilon \right ) - \frac 1{\varepsilon^{2}}W \left ( \frac {x+q} \varepsilon \right ) + o(1) \ \hbox{ as } \varepsilon \to 0 $$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.

We present in this paper the results of a research motivated by the need of a very fast solution of thermal flow in solar receivers. These receivers are composed by a large number of parallel pipes with the same geometry. We have introduced a reduced Schwarz algorithm that skips the computation in a large part of the pipes. The computation of the temperature in the skep domain is replaced by a reduced mapping that provides the transmission conditions. This reduced mapping is computed in an off-line stage. We have performed an error analysis of the reduced Schwarz algorithm, proving that the error is bounded in terms of the linearly decreasing error of the standard Schwarz algorithm, plus the error stemming from the reduction of the trace mapping. The last error is asymptotically dominant in the Schwarz iterative process. We obtain $L^2$ errors below $2\%$ with relatively small overlapping lengths., Comment: 18 pages, 7 figures

We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^5,~&\mbox{ in } \Omega \times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial \Omega \times \mathbb{R}^+,\\ u(x,0)=u_0(x),~&\mbox{ in } \Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^3$. Let $H_\gamma(x,y)$ be the regular part of the Green function of $-\Delta-\gamma$ in $\Omega$, where $\gamma \in (0,\lambda_1)$ and $\lambda_1$ is the first Dirichlet eigenvalue of $-\Delta$. Then, given a point $q\in \Omega$ such that $3\gamma(q)<\lambda_1$, where $$ \gamma(q)=\sup\{ \gamma>0: H_\gamma(q,q)>0 \}, $$ we prove the existence of a non-radial global positive and smooth solution $u(x,t)$ which blows up in infinite time with spike in $q$. The solution has the asymptotic profile $$ u(x,t)\sim 3^{\frac{1}{4}} \bigg(\frac{\mu(t)}{\mu(t)^2+|x-\xi(t)|^2}\bigg)^{\frac{1}{2}} \quad \text{as}\quad t \to \infty, $$ where $$ -\ln \mu(t)= 2\gamma(q) t(1+o(1)),\quad \xi(t)=q+O\big(\mu(t)\big) \quad \text{as}\quad t \to \infty. $$, Comment: 74 pages, 1 figure

We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$ \begin{cases} -\varepsilon^2 \Delta^{A} u=\frac{1}{2}(1-|u|^2)u,\\ \varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}. $$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2 minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$ u_\varepsilon(x)\to\frac{z}{|z|},\quad A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad x=\exp_y(z^\beta\nu_\beta(y)). $$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$. Here, $\{\nu_1,\nu_2\}$ is a normal frame for $M$ in $N$. This improves a recent result by De Philippis and Pigati who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense., Comment: 25 pages, 3 figures

A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid $3$-dimensional fluid. Helmholtz (1858) observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3d Euler equations exhibiting this motion pattern.

We consider the magnetic Ginzburg-Landau equations in $\mathbb{R}^4$ $$ \begin{cases} -\varepsilon^2(\nabla-iA)^2u = \frac{1}{2}(1-|u|^{2})u,\\ \varepsilon^2 d^*dA = \langle(\nabla-iA)u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{\mathbb{R}^4}|(\nabla-iA)u|^{2}+\varepsilon^2|dA|^{2}+\frac{1}{4\varepsilon^2}(1-|u|^{2})^{2}. $$ Here $u:\mathbb{R}^4\to \mathbb{C}$, $A: \mathbb{R}^4\to\mathbb{R}^4$ and $d$ denotes the exterior derivative acting on the one-form dual to $A$. Given a 2-dimensional minimal surface $M$ in $\mathbb{R}^3$ with finite total curvature and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ which has a zero set consisting of a smooth 2-dimensional surface close to $M\times \{0\}\subset \mathbb{R}^4$. Away from the latter surface we have $|u_\varepsilon| \to 1$ and $$ u_\varepsilon(x)\, \to\, \frac {z}{|z|},\quad A_\varepsilon(x)\, \to\, \frac 1{|z|^2} ( -z_2 \nu(y) + z_1 {\textbf{e}}_4), \quad x = y + z_1 \nu(y) + z_2 {\textbf{e}}_4 $$ for all sufficiently small $z\ne 0$. Here $y\in M$ and $\nu(y)$ is a unit normal vector field to $M$ in $\mathbb{R}^3$.

We consider the 3D Gross-Pitaevskii equation \begin{equation}\nonumber i\partial_t \psi +\Delta \psi+(1-|\psi|^2)\psi=0 \text{ for } \psi:\mathbb{R}\times \mathbb{R}^3 \rightarrow \mathbb{C} \end{equation} and construct traveling waves solutions to this equation. These are solutions of the form $\psi(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ of order $\varepsilon|\log\varepsilon|$ for a small parameter $\varepsilon>0$. We build two different types of solutions. For the first type, the functions $u$ have a zero-set (vortex set) close to an union of $n$ helices for $n\geq 2$ and near these helices $u$ has degree 1. For the second type, the functions $u$ have a vortex filament of degree $-1$ near the vertical axis $e_3$ and $n\geq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(\varepsilon\sqrt{|\log \varepsilon|)}$ from the axis and are solutions to the Klein-Majda-Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross-Pitaevskii equation, namely the Ginzburg-Landau equation. To prove the existence of these solutions we use the Lyapunov-Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.

For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t \theta+u\cdot \nabla \theta=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp \psi, \quad \psi = (-\Delta)^{-s}\theta \quad \text{in } \mathbb{R}^2 \times (0,T) , \end{aligned} \right. $$ $0

We consider the Euler equations in ${\mathbb R}^3$ expressed in vorticity form. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of {\em helical filaments}, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple similar helical filaments travelling and rotating together.

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=\Delta u+u^{3},~&\mbox{ in } \Omega\times (0,T),\\ u(x,t)=u|_{\partial\Omega},~&\mbox{ on } \partial\Omega\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \Omega, \end{cases} \end{equation*} where $5\leq n\leq 7$, $\Omega=\R^n$ or $\Omega \subset \R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {\em shrinking sphere} in $\Omega$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(\xi_r(t),\xi_z(t))}{\la(t)}\right|^2}$$ where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots,x_n)\in\Omega$. Moreover, the singularity location $$(\xi_r(t),\xi_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ for some fixed $z_0$, and the blow-up rate $$\la(t)\sim \frac{T-t}{|\log(T-t)|^2}~\mbox{ as }~t\nearrow T.$$ This is a completely new phenomenon in the parabolic setting., Comment: 64 pages; comments are welcome

We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^{5},~&\mbox{ in } \ {\mathbb R}^3 \times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \ {\mathbb R}^3. \end{cases} \end{equation*} We construct type II finite time blow-up solution $u(x,t)$ with the blow-up rates $ \| u\|_{L^\infty} \sim (T-t)^{-k}$, where $ k=1,2,... $. This gives a rigorous proof of the formal computations by Filippas, Herrero and Velazquez \cite{fhv}. This is the first instance of type II finite time blow-up for three dimensional energy critical heat equation., Comment: 43 pages; comments welcome

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; \Delta u - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v =&\; (-\Delta_{\R^2})^{-1} u := \frac 1{2\pi} \int_{R^2} \, \log \frac 1{|x-z|}\,u(z,t)\, dz, \\ & \qquad\ u(\cdot ,0) = u_0 \geq 0\quad\hbox{in } R^2. \end{aligned} \right. \end{equation} We consider the {\em critical mass case} $\int_{R^2} u_0(x)\, dx = 8\pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8\pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of \equ{ks0} is globally defined and blows-up in infinite time. As $t\to+\infty $ it has the approximate profile $$ u(x,t) \approx \frac 1{\lambda^2} \ch{U}\left (\frac {x-\xi(t)}{\lambda(t)} \right ), \quad \ch{U}(y)= \frac{8}{(1+|y|^2)^2}, $$ where $\lambda(t) \approx \frac c{\sqrt{\log t}}, \ \xi(t)\to q $ for some $c>0$ and $q\in \R^2$. This result answers affirmatively the nonradial stability conjecture raised in \cite{g}., Comment: 95 pages; final version; comments are welcome

We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional Laplace operator of order $2 s \in (0, 2)$ acting in one space dimension and the reaction is determined by a $1$-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of $N \ge 2$ equally oriented dislocations of size $1$. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When $s \in (1/2, 1)$, these solutions are shown to be asymptotically stable with respect to odd perturbations.

We consider the energy critical heat equation in $\mathbb R^n$ for $n\ge 7$ $$\left\{ \begin{aligned} u_t & = \Delta u+ |u|^{\frac 4{n-2}}u \hbox{ in }\ \mathbb R^n \times (0, \infty), \\ u(\cdot,0) & = u_0 \ \hbox{ in }\ \mathbb R^n, \end{aligned}\right. $$ which corresponds to the $L^2$-gradient flow of the Sobolev-critical energy $$ J(u) = \int_{\mathbb R^n} e[u] , \quad e[u] := \frac 12 |\nabla u|^2 - \frac {n-2}{2n} |u|^{\frac {2n}{n-2} }. $$ Given any $k\ge 2$ we find an initial condition $u_0$ that leads to sign-changing solutions with {\em multiple blow-up at a single point} (tower of bubbles) as $t\to +\infty$. It has the form of a superposition with alternate signs of singularly scaled {\em Aubin-Talenti solitons}, $$ u(x,t) = \sum_{j=1}^k (-1)^{j-1} {\mu_j^{-\frac {n-2}2}} U \left( \frac {x}{\mu_j} \right)\, +\, o(1) \quad\hbox{as } t\to +\infty $$ where $U(y)$ is the standard soliton $ U(y) = % (n(n-2))^{\frac 1{n-2}} \alpha_n\left ( \frac 1{1+|y|^2}\right)^{\frac{n-2}2}$ and $$\mu_j(t) = \beta_j t^{- \alpha_j}, \quad \alpha_j = \frac 12 \Big ( \, \left( \frac{n-2}{n-6}\right)^{j-1} -1 \Big). $$ Letting $\delta_0$ the Dirac mass, we have energy concentration of the form $$ e[ u(\cdot, t)]- e[U] \rightharpoonup (k-1) S_n\,\delta_{0} \quad\hbox{as } t\to +\infty $$ where $S_n=J(U)$. The initial condition can be chosen radial and compactly supported. We establish the codimension $k+ n (k-1)$ stability of this phenomenon for perturbations of the initial condition that have space decay $u_0(x) =O( |x|^{-\alpha})$, $\alpha > \frac {n-2}2$, which yields finite energy of the solution.

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} with $u(x,t): \bar \Omega\times [0,T) \to S^2$. Here $\Omega$ is a bounded, smooth axially symmetric domain in $\mathbb{R}^3$. We prove that for any circle $\Gamma \subset \Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $\Gamma$, in fact $$ |\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8\pi \delta_\Gamma \text{ as } t\to T . $$ for a regular function $u_*(x)$, where $\delta_\Gamma$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.

For each given $n\geq 2$, we construct a family of entire solutions $u_\varepsilon (z,t)$, $\varepsilon>0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation \begin{equation*}\nonumber \Delta u+(1-|u|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. \end{equation*} These solutions are $2\pi/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $\varepsilon\to 0$ $$ u_\varepsilon (z,t) \approx \prod_{j=1}^n W\left( z- \varepsilon^{-1} f_j(\varepsilon t) \right), $$ where $W(z) =w(r) e^{i\theta} $, $z= re^{i\theta},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ \Delta W+(1-|W|^2)W=0 \text{ in } \mathbb{R}^2 $ and $$ f_j(t) = \frac { \sqrt{n-1} e^{it}e^{2 i (j-1)\pi/ n }}{ \sqrt{|\log\varepsilon|}}, \quad j=1,\ldots, n. $$ Existence of these solutions was previously conjectured, being ${\bf f}(t) = (f_1(t),\ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ \mathcal W_\varepsilon ( {\bf f} ) :=\pi \int_0^{2\pi} \Big ( \, \frac{|\log \varepsilon|} 2 \sum_{k=1}^n|f'_k(t)|^2-\sum_{j\neq k}\log |f_j(t)-f_k(t)| \, \Big ) \mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ \lim_{|z| \to +\infty } |u_\varepsilon (z,t)| = 1 \quad \hbox{uniformly in $t$} $$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $\Omega$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in \Omega$ and $n\geq 5$. We prove that there exists an integer $k_0$ such that for any integer $k\geq k_0$ there exist initial data $u_0$ and smooth parameter functions $\xi(t)\to q$, $0<\mu(t)\to 0$ as $t\to +\infty$ such that the solution $u_q$ of the critical nonlinear heat equation \begin{equation*} \begin{cases} u_t = \Delta u + |u|^{\frac{4}{n-2}}u\text{ in } \Omega\times (0, \infty),\\ u = 0\text{ on } \partial \Omega\times (0, \infty),\\ u(\cdot, 0) = u_0 \text{ in }\Omega, \end{cases} \end{equation*} has the form \begin{equation*} u_q(x, t) \approx \mu(t)^{-\frac{n-2}{2}}\left(Q_k\left(\frac{x-\xi(t)}{\mu(t)}\right) - H(x, q)\right), \end{equation*} where the profile $Q_k$ is the non-radial sign-changing solution of the Yamabe equation \begin{equation*} \Delta Q + |Q|^{\frac{4}{n-2}}Q = 0\text{ in }\mathbb{R}^n, \end{equation*} constructed in \cite{delpinomussofrankpistoiajde2011}. In dimension 5 and 6, we also prove the stability of $u_q(x, t)$., Comment: 60 pages; comments welcome

We consider the Cauchy problem for the energy critical heat equation $$ u_t = \Delta u + |u|^{\frac 4{n-2}}u {{\quad\hbox{in } }} \ {\mathbb R}^n \times (0, T), \quad u(\cdot,0) =u_0 {{\quad\hbox{in } }} {\mathbb R}^n $$ in dimension $n=5$. More precisely we find that for given points $q_1, q_2,\ldots, q_k$ and any sufficiently small $T>0$ there is an initial condition $u_0$ such that the solution $u(x,t)$ of the problem blows-up at exactly those $k$ points with rates type II, namely with absolute size $ \sim (T-t)^{-\alpha} $ for $\alpha > \frac 34 $. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles., Comment: 15 pages

We consider the wave equation $\varepsilon^2(-\partial_t^2 + \Delta)u + f(u) = 0$ for $0<\varepsilon\ll 1$, where $f$ is the derivative of a balanced, double-well potential, the model case being $f(u) = u-u^3$. For equations of this form, we construct solutions that exhibit an interface of thickness $O(\varepsilon )$ that separates regions where the solution is $O(\varepsilon^k)$ close to $\pm 1$, and that is close to a timelike hypersurface of vanishing {\em Minkowskian} mean curvature. This provides a Minkowskian analog of the numerous results that connect the Euclidean Allen-Cahn equation and minimal surfaces or the parabolic Allen-Cahn equation and motion by mean curvature. Compared to earlier results of the same character, we develop a new constructive approach that applies to a larger class of nonlinearities and yields much more precise information about the solutions under consideration., Comment: 34 pages

A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. is that of finding regular solutions with highly concentrated vorticities around $N$ moving {\em vortices}. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. In this paper we devise a {\em gluing approach} for the construction of smooth $N$-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville's equation plus small, more regular terms. Gluing methods have been a powerful tool in geometric constructions by {\em desingularization}. We succeed in applying those ideas in this highly challenging setting.

We consider the problem v_t & = \Delta v+ |v|^{p-1}v \quad\hbox{in }\ \Omega\times (0, T), v & =0 \quad\hbox{on } \partial \Omega\times (0, T ) , v& >0 \quad\hbox{in }\ \Omega\times (0, T) . In a domain $\Omega\subset \mathbb R^d$, $d\ge 7$ enjoying special symmetries, we find the first example of a solution with type II blow-up for a power $p$ less than the Joseph-Lundgren exponent $$p_{JL}(d)=\infty, & \text{if $3\le d\le 10$}, 1+{4\over d-4-2\,\sqrt{d-1}}, & \text{if $d\ge11$}. $$ No type II radial blow-up is present for $p< p_{JL}(d)$. We take $p=\frac{d+1}{d-3}$, the Sobolev critical exponent in one dimension less. The solution blows up on circle contained in a negatively curved part of the boundary in the form of a sharply scaled Aubin-Talenti bubble, approaching its energy density a Dirac measure for the curve. This is a completely new phenomenon for a diffusion setting., Comment: 49 pages, no figures. arXiv admin note: text overlap with arXiv:1705.01672

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension three $$ u_t = \Delta u + u^5 , \quad {\mbox {in}} \quad \R^3 \times (0,\infty), \ \ u(x, 0)= u_0 (x)\inn \R^3. $$ For each $\gamma>1$ we find initial data (not necessarily radially symmetric) with $\lim\limits_{r \to \infty} |x|^\gamma u_0 (x) >0$ such that as $t \to \infty$ $$ \| u(\cdot ,t ) \|_\infty \sim t^{\gamma-1 \over 2} , \quad {\mbox {if}} \quad 1<\gamma <2, \quad \| u(\cdot ,t ) \|_\infty \sim \sqrt{t}, \quad {\mbox {if}} \quad \gamma >2, \quad $$ and $$ \| u(\cdot , t)\|_\infty \sim \sqrt{t}\, (\ln t )^{-1} , \quad {\mbox {if}} \quad \gamma = 2. $$ Furthermore we show that this infinite time blow-up is co-dimensional one stable. The existence of such solutions was conjectured by Fila and King.

We consider the parabolic Allen-Cahn equation in $\mathbb{R}^n$, $n\ge 2$, $$u_t= \Delta u + (1-u^2)u \quad \hbox{ in } \mathbb{R}^n \times (-\infty, 0].$$ We construct an ancient radially symmetric solution $u(x,t)$ with any given number $k$ of transition layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling copies of $w$ with spherical interfaces distant $O(\log |t| )$ one to each other as $t\to -\infty$. These interfaces are resemble at main order copies of the {\em shrinking sphere} ancient solution to mean the flow by mean curvature of surfaces: $|x| = \sqrt{- 2(n-1)t}$. More precisely, if $w(s)$ denotes the heteroclinic 1-dimensional solution of $w'' + (1-w^2)w=0$ $w(\pm \infty)= \pm 1$ given by $w(s) = \tanh \left(\frac s{\sqrt{2}} \right) $ we have $$ u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(|x|-\rho_j(t)) - \frac 12 (1+ (-1)^{k}) \quad \hbox{ as } t\to -\infty $$ where $$\rho_j(t)=\sqrt{-2(n-1)t}+\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log\left(\frac {|t|}{\log |t| }\right)+ O(1),\quad j=1,\ldots ,k.$$

We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+ u(1-u^2)\quad (x,t)\in \mathbb{R}\times (-\infty, 0].$$ The steady state $w(x) =\tanh (x/\sqrt{2})$, connects, as a "transition layer" the stable phases $-1$ and $+1$. We construct a solution $u$ with any given number $k$ of transition layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling copies of $w$ with interfaces diverging one to each other as $t\to -\infty$. More precisely, we find $$ u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(x-\xi_j(t)) + \frac 12 ((-1)^{k-1}- 1)\quad \hbox{as} t\to -\infty, $$ where the functions $\xi_j(t)$ satisfy a first order Toda-type system. They are given by $$\xi_j(t)=\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log(-t)+\gamma_{jk},\quad j=1,...,k,$$ for certain explicit constants $\gamma_{jk}.$

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} where $\Omega$ is a bounded, smooth domain in $\mathbb{R}^2$, $u: \Omega\times(0,T)\to S^2$, $u_0:\bar\Omega \to S^2$ is smooth, and $\varphi = u_0\big|_{\partial\Omega}$. Given any points $q_1,\ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corrotational harmonic map. We build a continuation after blow-up as a $H^1$-weak solution with a finite number of discontinuities in space-time by "reverse bubbling", which preserves the homotopy class of the solution after blow-up., Comment: 76 pages. Final version

Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \Delta u + u^{\frac{n+2}{n-2}} \inn \Omega\times (0,\infty), \quad u =0 \onn \pp\Omega\times (0,\infty). $$ Let $G(x,y)$ be the Dirichlet Green's function of $-\Delta$ in $\Omega$ and $H(x,y)$ its regular part. Let $q_j\in \Omega$, $j=1,\ldots,k$, be points such that the matrix $$ \left [ \begin{matrix} H(q_1, q_1) & -G(q_1,q_2) &\cdots & -G(q_1, q_k) -G(q_1,q_2) & H(q_2,q_2) & -G(q_2,q_3) \cdots & -G(q_3,q_k) \vdots & & \ddots& \vdots -G(q_1,q_k) &\cdots& -G(q_{k-1}, q_k) & H(q_k,q_k) \end{matrix} \right ] $$ is positive definite. For any $k\ge 1$ such points indeed exist. We prove the existence of a positive smooth solution $u(x,t)$ which blows-up by bubbling in infinite time near those points. More precisely, for large time $t$, $u$ takes the approximate form $$ u(x,t) \approx \sum_{j=1}^k \alpha_n \left ( \frac { \mu_j(t)} { \mu_j(t)^2 + |x-\xi_j(t)|^2 } \right )^{\frac {n-2}2} . $$ Here $\xi_j(t) \to q_j$ and $0<\mu_j(t) \to 0$, as $t \to \infty$. We find that $\mu_j(t) \sim t^{-\frac 1{n-4}} $ as $t\to +\infty$, when $n\geq 5$.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions., Comment: arXiv admin note: substantial text overlap with arXiv:1509.08803

We consider the problem of finding positive solutions of the problem $\Delta u - \lambda u +u^5 = 0$ in a bounded, smooth domain $\Omega$ in $\mathbb{R}^3$, under zero Neumann boundary conditions. Here $\lambda$ is a positive number. We analyze the role of Green's function of $-\Delta +\lambda$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $\lambda$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value $\lambda_*$ such that if $\lambda-\lambda^*>0$ is sufficiently small, then this problem is solvable by a solution $u_\lambda$ which blows-up by bubbling at a certain interior point of $\Omega$ as $\lambda \downarrow \lambda_*$., Comment: 26 pages, 1 figure

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in $\mathbb R^3$, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $\alpha^2 \Delta u + u(1-u^2)=0, \quad \hbox{in }\Omega\subset \R^N $ where $N=3$, $\Omega$ is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $\alpha \to 0$, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally $\partial \Omega$ of the domain and that is non-degenerate respect to $\Omega$. We provide explicit examples of surfaces to which our result applies., Comment: 30 pages

We construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the singularity. We find in $\mathbb R^3$ a surface $M$ orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface $\Sigma(t) = M + te_z$ evolves by mean curvature flow. This amounts to the equation $H_M = \nu\cdot e_z$ where $H_M$ denotes mean curvature, $\nu$ is a choice of unit normal to $M$, and $e_z$ is a unit vector along the $z$-axis. The surface $M$ is in correspondence with the classical 3-end Costa-Hoffmann-Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded, complete minimal surfaces with finite total curvature.

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $$f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0. $$ Let $p_1,\ldots,p_n$ be any set of points at which $f(p_i)=0$ and $D^2f(p_i)$ is non-singular. We prove that for all sufficiently small $\lambda>0$ there exists a family of "bubbling" conformal metrics $g_\lambda=e^{u_\lambda}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_\lambda}=-f+\lambda^2$. Moreover, the family $u_\lambda$ satisfies $$u_\lambda(p_j) = -4\log\lambda -2\log \left (\frac 1{\sqrt{2}} \log \frac 1\lambda \right ) +O(1)$$ and $$\lambda^2e^{u_\lambda}\rightharpoonup8\pi\sum_{i=1}^{n}\delta_{p_i},\quad \mbox{as }\lambda \to 0,$$ where $\delta_{p}$ designates Dirac mass at the point $p$., Comment: 29 pages, 1 figure

For a smooth, bounded domain $\Omega$, $s\in(0,1)$, $p\in \left(1,\frac{n+2s}{n-2s}\right)$ we consider the nonlocal equation $$ \epsilon^{2s} (-\Delta)^s u+u=u^p \quad {\mbox{in}}\Omega $$ with zero Dirichlet datum and a small parameter $\epsilon>0$. We construct a family of solutions that concentrate as $\epsilon \to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function of $ \epsilon^{2s} (-\Delta)^s +1$ in the expanding domain $\epsilon^{-1}\Omega$ with zero exterior datum.

We consider the boundary value problem $-\Delta u + u =\lambda e^u$ in $\Omega$ with Neumann boundary condition, where $\Omega$ is a bounded smooth domain in $\mathbb R^2$, $\lambda>0.$ This problem is equivalent to the stationary Keller-Segel system from chemotaxis. We establish the existence of a solution $u_\lambda$ which exhibits a sharp boundary layer along the entire boundary $\partial\Omega$ as $\lambda\to 0$. These solutions have large mass in the sense that $ \int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.$

The nonlocal $s$-fractional minimal surface equation for $\Sigma= \partial E$ where $E$ is an open set in $R^N$ is given by $$ H_\Sigma^ s (p) := \int_{R^N} \frac {\chi_E(x) - \chi_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in \Sigma. $$ Here $0