arXiv subjects

Pierre Raphael

Publications and source records attributed to Pierre Raphael.

At least 19 recordsLinked to original sources

arXiv2020

On weakly turbulent solutions to the perturbed linear Harmonic oscillator

We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.

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arXiv2019

On smooth self similar solutions to the compressible Euler equations

We consider the barotropic Euler equations in dimension d>1 with decaying density at spatial infinity. The phase portrait of the nonlinear ode governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated light (acoustic) cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic C^\infty self-similar solutions with suitable decay at infinity. The C^\infty regularity is used in a fundamental way in the companion papers \cite{MRRSnls}, \cite{MRRSfluid} to control the associated linearized operator, and construct finite energy blow up solutions of respectively the defocusing nonlinear Schr\"odinger equation in dimension $5\le d\le9$, and the isentropic ideal compressible Euler and Navier-Stokes equations in dimensions d=2,3.

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arXiv2019

On blow up for the energy super critical defocusing non linear Schr\"odinger equations

We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal C^\infty$ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression in the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $\mathcal C^\infty$ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper \cite{MRRSprofile}.

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arXiv2019

On the implosion of a three dimensional compressible fluid

We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. Two essential steps of the analysis are the existence of $\mathcal C^\infty$ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow which are addressed in the companion papers \cite{MRRSprofile, MRRSdefoc}. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

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arXiv2017

On strongly anisotropic type I blow up

We consider the energy super critical 4 dimensional semilinear heat equation $$\partial_tu=\Delta u+|u|^{p-1}u, \ \ x\in \Bbb R^4, \ \ p>5.$$ Let $\Phi(r)$ be a three dimensional radial self similar solution for the three supercritical probmem as exhibited and studied in \cite{CRS}. We show the finite codimensional transversal stability of the corresponding blow up solution by exhibiting a manifold of finite energy blow up solutions of the four dimensional problem with cylindrical symmetry which blows up as $$u(t,x)\sim \frac{1}{(T-t)^{\frac{1}{p-1}}}U(t,Y), \ \ Y=\frac{x}{\sqrt{T-t}}$$ with the profile $U$ given to leading order by $$U(t,Y)\sim\frac{1}{(1+b(t)z^2)^{\frac 1{p-1}}}\Phi\left(\frac{r}{\sqrt{1+b(t)z^2}}\right), \ \ Y=(r,z), \ \ b(t)=\frac{c}{|\log(T-t)|}$$ corresponding to a constant profile $\Phi(r)$ in the $z$ direction reconnected to zero along the moving free boundary $|z(t)|\sim \frac{1}{\sqrt{b}}\sim \sqrt{|\log (T-t)|}.$ Our analysis revisits the stability analysis of the self similar ODE blow up \cite{BK, MZduke,MZgaffa} and combines it with the study of the Type I self similar blow up \cite{CRS}. This provides a robust canonical framework for the construction of strongly anisotropic blow up bubbles.

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arXiv2017

On strongly anisotropic type II blow up

We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\partial_tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ A fundamental open problem on this canonical nonlinear model is to understand the possible blow up profiles appearing after renormalization of a singularity. We exhibit in this paper a new scenario corresponding to the first example of strongly anisotropic blow up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve the {\it reconnection problem} in order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilization of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow up speeds.

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arXiv2016

On the stability of type I blow up for the energy super critical heat equation

We consider the energy super critical semilinear heat equation $$\partial_t u=\Delta u+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.$$ We first revisit the construction of radially symmetric backward self similar solutions and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.

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arXiv2016

Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions

We consider the energy critical semilinear heat equation $$\partial_tu=\Delta u+|u|^{\frac{4}{d-2}}u, \ \ x\in \mathbb R^d$$ and give a complete classification of the flow near the ground state solitary wave $$Q(x)=\frac{1}{\left( 1+\frac{|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$ in dimension $d\ge 7$, in the energy critical topology and without radial symmetry assumption. Given an initial data $Q+\varepsilon_0$ with $\parallel \nabla \varepsilon_0\parallel_{L^2}\ll 1$, the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension $d\ge 7$ near the solitary wave even though it is known to occur in smaller dimensions. Our proof is based on sole energy estimates deeply and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around $Q$ belonging to the unstable manifold.

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arXiv2015

Strongly interacting blow up bubbles for the mass critical NLS

We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conformal one. Such solution concentrates K bubbles at a point. These special behaviors are due to strong interactions between the waves, in contrast with previous works on multi-solitary waves of (NLS) where interactions do not affect the blow up rate.

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arXiv2015

On melting and freezing for the 2d radial Stefan problem

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball $\{r\leq \lambda(t)\}$. We revisit the pioneering analysis of [20] and prove the existence in the radial class of finite time melting regimes $$ \lambda(t)=\left\{\begin{array}{ll} (T-t)^{1/2}e^{-\frac{\sqrt{2}}{2}\sqrt{|\ln(T-t)|}+O(1)}\\ (c+o(1))\frac{(T-t)^{\frac{k+1}{2}}}{|\ln (T-t)|^{\frac{k+1}{2k}}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to T $$ which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k\in \Bbb N^*$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [42] by introducing a new and canonical functional framework for the study of type II (i.e. non self similar) blow up. We also show a deep duality between the construction of the melting regimes and the derivation of a discrete sequence of global-in-time freezing regimes $$ \lambda_\infty - \lambda(t)\sim\left\{\begin{array}{ll} \frac{1}{\log t}\\ \frac{1}{t^{k}(\log t)^{2}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to +\infty $$ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable.

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arXiv2015

Codimension one threshold manifold for the critical gKdV equation

We construct the 'threshold manifold' near the soliton for the mass critical gKdV equation, completing results obtained in arXiv:1204.4625 and arXiv:1204.4624. In a neighborhood of the soliton, this C1 manifold of codimension one separates solutions blowing up in finite time and solutions in the 'exit regime'. On the manifold, solutions are global in time and converge locally to a soliton. In particular, the soliton behavior is strongly unstable by blowup.

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arXiv2014

Near soliton dynamics and singularity formation for $L^2$ critical problems

This survey reviews the state of the art concerning the singularity formation for two canonical dispersive problems: the mass critical non linear Schr\"odinger equation and the mass critical generalized KdV equation. In particular, we address the question of the classification of the flow for initial data near the soliton.

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arXiv2014

Type II blow up for the energy supercritical NLS

We consider the energy super critical nonlinear Schrödinger equation $$i\pa_tu+Δu+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation, which was done using maximum principle techniques following. Instead we develop a robust energy method, in continuation of the works in the energy and mass critical cases. This shades a new light on the essential role played by the solitary wave and its tail in the type II blow up mechanism, and the universality of the corresponding singularity formation in both energy critical and super critical regimes.

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arXiv2014

Minimal mass blow up solutions for a double power nonlinear Schrödinger equation

We consider a nonlinear Schrödinger equation with double power nonlinearity, where one power is focusing and mass critical and the other mass sub-critical. Classical variational arguments ensure that initial data with mass less than the mass of the ground state of the mass critical problem lead to global in time solutions. We are interested by the threshold dynamic and in particular by the existence of finite time blow up minimal solutions. For the mass critical problem, such an object exists thanks to the explicit conformal symmetry, and is in fact unique. For the focusing double power nonlinearity, we exhibit a new class of minimal blow up solutions with blow up rates deeply affected by the double power nonlinearity. The analysis adapts the recent approach developed by Raphaël and Szeftel for the construction of minimal blow up elements.

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arXiv2013

Quantized slow blow up dynamics for the corotational energy critical harmonic heat flow

We consider the energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem $$\partial_t u -\pa^2_{r} u-\frac{\pa_r u}{r} + \frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$ . Given an integer $L\in \Bbb N^*$, we exhibit a set of initial data arbitrarily close to the least energy harmonic map $Q$ in the energy critical topology such that the corresponding solution blows up in finite time by concentrating its energy $$\nabla u(t,r)-\nabla Q(\frac{r}{\l(t)})\to u^* in L^2$$ at a speed given by the {\it quantized} rates: $$\l(t)=c(u_0)(1+o(1))\frac{(T-t)^L}{|\log (T-t)|^{\frac{2L}{2L-1}}},$$ in accordance with the formal predictions [3]. The case L=1 corresponds to the stable regime exhibited in [37], and the data for $L\ge 2$ leave on a manifold of codimension $(L-1)$ in some weak sense. Our analysis lies in the continuation of [36,32,37] by further exhibiting the mechanism for the existence of the excited slow blow up rates and the associated instability of these threshold dynamics.

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arXiv2012

Blow up for the critical gKdV equation III: exotic regimes

We consider the blow up problem in the energy space for the critical (gKdV) equation in the continuation of part I and part II. We know from part I that the unique and stable blow up rate for solutions close to the solitons with strong decay on the right is $1/t$. In this paper, we construct non-generic blow up regimes in the energy space by considering initial data with explicit slow decay on the right in space. We obtain finite time blow up solutions with speed $t^{-ν}$ where $ ν>11/13,$ as well as global in time growing up solutions with both exponential growth or power growth. These solutions can be taken with initial data arbitrarily close to the ground state solitary wave.

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arXiv2012

On the stability of critical chemotactic aggregation

We consider the two dimensional parabolic-elliptic Patlak-Keller-Segel model of chemotactic aggregation for radially symmetric initial data. We show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process.

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arXiv2012

Blow up for the critical gKdV equation II: minimal mass dynamics

We fully revisit the near soliton dynamics for the mass critical (gKdV) equation. In Part I, for a class of initial data close to the soliton, we prove that only three scenario can occur: (BLOW UP) the solution blows up in finite time $T$ in a universal regime with speed $1/(T-t)$; (SOLITON) the solution is global and converges to a soliton in large time; (EXIT) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant $L^2$ norm. Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this class that any nonpositive energy initial data (except solitons) yields finite time blow up, thus obtaining the classification of the solitary wave at zero energy. In Part II, we classify minimal mass blow up by proving existence and uniqueness (up to invariances of the equation) of a minimal mass blow up solution $S(t)$. We also completely describe the blow up behavior of $S(t)$. Second, we prove that $S(t)$ is the universal attractor in the (EXIT) case, i.e. any solution as above in the (EXIT) case is close to $S$ (up to invariances) in $L^2$ at the exit time. In particular, assuming scattering for $S(t)$ (in large positive time), we obtain that any solution in the (EXIT) scenario also scatters, thus achieving the description of the near soliton dynamics.

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