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Pierre Raphael

Publications and source records attributed to Pierre Raphael.

31 records · Page 2Linked to original sources
arXiv2012

Blow up for the critical gKdV equation I: dynamics near the soliton

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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arXiv2012

Nondispersive solutions to the L2-critical half-wave equation

We consider the focusing $L^2$-critical half-wave equation in one space dimension $$ i \partial_t u = D u - |u|^2 u, $$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u \|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2} \geq M_*$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.

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arXiv2011

Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow

We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$ which reduces to the semilinear 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 $f$. The corresponding initial data can be chosen smooth, well localized and arbitrarily close to the ground state harmonic map in the energy critical topology. We give sharp asymptotics on the corresponding singularity formation which occurs through the concentration of a universal bubble of energy at the speed predicted in [Van den Bergh, J.; Hulshof, J.; King, J., Formal asymptotics of bubbling in the harmonic map heat flow, SIAM J. Appl. Math. vol 63, o5. pp 1682-1717]. Our approach lies in the continuation of the study of the 1-equivariant energy critical wave map and Schrödinger map with $\Bbb S^2$ target in [Raphaël, P.; Rodnianksi, I., Stable blow up dynamics for the critical corotational wave maps and equivariant Yang Mills problems, to appear in Prep. Math. IHES.], [Merle, F.; Raphaël, P.; Rodnianski, I., Blow up dynamics for smooth solutions to the energy critical Schrödinger map, preprint 2011.].

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arXiv2011

Blow up dynamics for smooth data equivariant solutions to the energy critical Schrodinger map problem

We consider the energy critical Schrodinger map to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map in the scale invariant norm which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. The concentration rate is given by $$λ(t)=κ(u)\frac{T-t}{|\log (T-t)|^2}(1+o(1))$$ for some $κ(u)>0$. The detailed proofs of the results will appear in a companion paper.

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arXiv2010

The instability of Bourgain-Wang solutions for the L^2 critical NLS

We consider the two dimensional $L^2$ critical nonlinear Schrödinger equation $i\pa_tu+Δu+u|u|^2=0$. In the pioneering work \cite{BW}, Bourgain and Wang have constructed smooth solutions which blow up in finite time $T<+\infty$ with the pseudo conformal speed $$\|\nabla u(t)\|_{L^2}\sim \frac{1}{T-t},$$ and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical $L^2$ mass lies on the boundary of both $H^1$ open sets of global solutions that scatter forward and backwards in time, and solutions that blow up in finite time on the right in the log-log regime exhibited in \cite{MR1}, \cite{MR4}, \cite{R1}. We moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in \cite{Mcpam} in the critical mass case

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arXiv2010

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov in 1961. In a previous work (arXiv:0904.2443), we derived the stability of anisotropic models under {\it spherically symmetric perturbations} using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature. In this work, we show how this approach combined with a {\it new generalized} Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.

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arXiv2010

A new variational approach to the stability of gravitational systems

We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the flow. This was proved at the linear level by several authors based on the pioneering work by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this paper, inspired by pioneering works from the physics litterature (Lynden-Bell 94, Wiechen-Ziegler-Schindler MNRAS 88, Aly MNRAS 89), we use the monotonicity of the Hamiltonian under generalized symmetric rearrangement transformations to prove that non increasing steady solutions are local minimizer of the Hamiltonian under equimeasurable constraints, and extract compactness from suitable minimizing sequences. This implies the nonlinear stability of nonincreasing anisotropic steady states under radially symmetric perturbations.

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arXiv2010

Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+Δu+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2} M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow up elements at a non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$.

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arXiv2009

Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations

We consider the focusing nonlinear Schrödinger equations $i\partial_t u+Δu +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up time

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arXiv2009

Stable ground states for the relativistic gravitational Vlasov-Poisson system

We consider the three dimensional gravitational Vlasov-Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers --which is mostly unknown-- but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.

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arXiv2008

On Stability of Pseudo-Conformal Blowup for L^2-critical Hartree NLS

We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (Φ\ast |u|^2 \big) u \quad {in $\RR^4$},$$ where $Φ(x)$ is a perturbation of the convolution kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions $u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \| \nabla u(t) \|_{L^2_x} \sim \frac{1}{|t|} \quad {as} \quad t \nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by extending the nonlinear wave operator construction by Bourgain and Wang (see \cite{Bourgain+Wang1997}) to $L^2$-critical Hartree NLS.

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arXiv2007

Blow up of the critical norm for some radial L^2 super critical nonlinear Schrodinger equations

We consider the nonlinear Schrödinger equation $iu_t=-Δu-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}<p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is $\dot{H}^{s_c}$ with $0<s_c<1$ and this covers the physically relevant case $N=p=3$. The existence of finite time blow up solutions is known. Let $u(t)\in \dot{H}^{s_c}\cap \dot{H}^1$ be a radially symmetric blow up solution which blows up at $0<T<+\infty$, we prove that the scaling invariant $L^{p_c}$ norm where $\dot{H}^{s_c}\rightharpoonup L^{p_c}$ also blows up with a lower bound $|u(t)|_{L^{p_c}}\geq |\log(T-t)|^{C_{N,p}} $ as $t\to T$.

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