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Monica Musso

Publications and source records attributed to Monica Musso.

At least 37 records · Page 2Linked to original sources
arXiv2024

Overhanging solitary water waves

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.

math.AP↗
arXiv2024

Asymptotic properties of vortex-pair solutions for incompressible Euler equations in $\mathbb{R}^2$

A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ ω_t + \nabla^\perp Ψ\cdot \nabla ω= 0 , \quad Ψ= (-Δ)^{-1} ω, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $ω(x,t) = W(x_1-ct,x_2 )$ 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.

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arXiv2023

Nodal cluster solutions for the Brezis-Nirenberg problem in dimensions $N\geq 7$

We show that the classical Brezis-Nirenberg problem $$Δu + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad Ω, \quad u= 0 , \quad {\mbox {on}} \quad \partial Ω$$ admits nodal solutions clustering around a point on the boundary of $Ω$ as $\varepsilon \to 0$, for smooth bounded domains $Ω\subset \mathbb{R}^N $ in dimensions $N\geq 7$.

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arXiv2023

Existence of finite time blow-up in Keller-Segel system

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =Δu - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-Δ_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2π} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8π<m\le 8π+\varepsilon$ and any $k$ given points $q_{1},...,q_{k}$ in $\mathbb{R}^{2}$ there is an initial data $u_0^*$ of $(\star)$ for which the solution $u(x,t)$ blows-up in finite time as $t\to T$ with the approximate profile $$u(x,t)=\sum_{j=1}^{k}\frac{1}{λ_{j}^{2}(t)}U\left(\frac{x-ξ_{j}(t)}{λ_{j}(t)}\right)(1+o(1)), U(y)=\frac{8}{(1+|y|^{2})^{2}},$$ with $λ_{j}(t) \approx 2e^{-\frac{γ+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}} $ where $γ=0.57721...$ is the Euler-Mascheroni constant, $ξ_{j}(t)\to q_{j}\in \mathbb{R}^{2}$ and such that $\int_{\mathbb{R}^2}u(x,t)dx=km.$ This construction generalizes the existence result of the stable blow-up dynamics recently proved in \cite{CGMN1,CGMN2}.

math.AP↗
arXiv2023

Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

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.

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arXiv2023

Existence and stability of infinite time blow-up in the Keller-Segel system

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; Δu - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v =&\; (-Δ_{\R^2})^{-1} u := \frac 1{2π} \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} $\int_{R^2} u_0(x)\, dx = 8π$ 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 $8π$ 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 $$ u(x,t) \approx \frac 1{λ^2} \ch{U}\left (\frac {x-ξ(t)}{λ(t)} \right ), \quad \ch{U}(y)= \frac{8}{(1+|y|^2)^2}, $$ where $λ(t) \approx \frac c{\sqrt{\log t}}, \ ξ(t)\to q $ for some $c>0$ and $q\in \R^2$. This result answers affirmatively the nonradial stability conjecture raised in \cite{g}.

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arXiv2022

Doubling the equatorial for the prescribed scalar curvature problem on ${\mathbb{S}}^N$

We consider the prescribed scalar curvature problem on $ {\mathbb{S}}^N $ $$ Δ_{{\mathbb S}^N} v-\frac{N(N-2)}{2} v+\tilde{K}(y) v^{\frac{N+2}{N-2}}=0 \quad \mbox{on} \ {\mathbb S}^N, \qquad v >0 \quad \mbox{on} \ {\mathbb S}^N, $$ under the assumptions that the scalar curvature $\tilde K$ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of $O(3)$ obtained doubling the equatorial. We use the finite dimensional Lyapunov-Schmidt reduction method.

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arXiv2021

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension $N \ge 3$ is known to exist for all time $t$ and converges to a solution to the Yamabe problem as $t \to \infty$. We prove that if a suitable perturbation, which may be smooth and arbitrarily small, is imposed on the Yamabe flow on any given Riemannian manifold $M$ of dimension $N \ge 5$, the resulting flow may blow up at multiple points on $M$ in the infinite time. Our proof is constructive, and indeed we construct such a flow by using solutions of the Yamabe problem on the unit sphere $\mathbb{S}^N$ as blow-up profiles. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

math.AP↗
arXiv2021

Non-degeneracy and existence of new solutions for the Schrödinger equations

We consider the following nonlinear problem $$ (P) \quad \quad - Δu + V(|y|)u=u^{p},\quad u>0 \quad \mbox{in} \ {\mathbb{R}}^N, \quad u \in H^1({\mathbb{R}}^N), $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that the multi-bump solutions constructed in [20] is non-degenerate in a suitable symmetric space. We also use this non-degenerate result to construct new solutions for (P).

math.AP↗
arXiv2021

Doubling nodal solutions to the Yamabe equation in $\mathbb{R}^n$ with maximal rank

We construct a new family of entire solutions to the Yamabe equation $$-Δu=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u \mbox{ in }\mathcal{D}^{1,2}(\mathbb{R}^n).$$ If $n=3$, our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in $S^3(1)$.

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arXiv2020

On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit

In Euclidean 3-space endowed with a Cartesian reference system we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size $a$ and $n$ lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when $n$ is large. Considering a class of mappings $H\colon\mathbb{R}^{3}\to\mathbb{R}$ such that $H(X)\to 1$ as $|X|\to\infty$ with some decay of inverse-power type, we show that for $n$ large and $|a|$ small, in a suitable neighborhood of any Delaunay torus with $n$ lobes and neck-size $a$ there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals $H$ at every point.

math.AP↗
arXiv2020

Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t θ+u\cdot \nabla θ=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp ψ, \quad ψ= (-Δ)^{-s}θ\quad \text{in } \mathbb{R}^2 \times (0,T) , \end{aligned} \right. $$ $0<s<1$, we consider for $k\ge1$ the problem of finding a family of $k$-vortex solutions $θ_\varepsilon(x,t)$ such that as $\varepsilon\to 0$ $$ θ_\varepsilon(x,t) \rightharpoonup \sum_{j=1}^k m_jδ(x-ξ_j(t)) $$ for suitable trajectories for the vortices $x=ξ_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-Δ)^sW = (W-1)^γ_+, \quad \text{in } \mathbb{R}^2, \quad 1<γ< \frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in \cite{chan_uniqueness_2020}.

math.AP↗
arXiv2020

Travelling helices and the vortex filament conjecture in the incompressible Euler equations

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.

math.AP↗
arXiv2020

New type of solutions for the Nonlinear Schrödinger Equation in $\mathbb{R}^N$

We construct a new family of entire solutions for the nonlinear Schrödinger equation \begin{align*} \begin{cases} -Δu+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases} \end{align*} where $p\in (1, \frac{N+2}{N-2})$ and $N\geq 3$, and $V (y)= V(|y|)$ is a positive bounded radial potential satisfying $$ V(|y|) = V_0 + \frac{a}{|y|^m} + O( \frac{1}{|y|^{m+σ}} ), \quad {\mbox {as}} \quad |y| \to \infty , $$ for some fixed constants $V_0, a, σ>0$, and $m>1$. Our solutions have strong analogies with the doubling construction of entire finite energy sign-changing solution for the Yamabe equation.

math.AP↗
arXiv2020

New type II Finite time blow-up for the energy supercritical heat equation

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=Δu+u^{3},~&\mbox{ in } Ω\times (0,T),\\ u(x,t)=u|_{\partialΩ},~&\mbox{ on } \partialΩ\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $5\leq n\leq 7$, $Ω=\R^n$ or $Ω\subset \R^n$ 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 $Ω$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(ξ_r(t),ξ_z(t))}{\la(t)}\right|^2}$$ where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots,x_n)\inΩ$. Moreover, the singularity location $$(ξ_r(t),ξ_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ 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.

math.AP↗
arXiv2020

Type II Finite time blow-up for the three dimensional energy critical heat equation

We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+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.

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