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Frank Merle

Publications and source records attributed to Frank Merle.

At least 37 records · Page 2Linked to original sources
arXiv2017

On strongly anisotropic type II blow up

We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\partial_tu=Δu+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ 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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arXiv2017

Inelasticity of soliton collisions for the 5D energy critical wave equation

For the focusing energy critical wave equation in 5D, we construct a solution showing the inelastic nature of the collision of any two solitons, except the special case of two solitons of same scaling and opposite signs. Beyond its own interest as one of the first rigorous studies of the collision of solitons for a non-integrable model, the case of the quartic gKdV equation being partially treated by the authors in previous works, this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation. This conjecture has already been established in the 3D radial case and in the general case in 3, 4 and 5D along a sequence of times by Duyckaerts, Kenig and Merle. The study of the nature of the collision requires a refined approximate solution of the two-soliton problem and a precise determination of its space asymptotics. To prove inelasticity, these asymptotics are combined with the method of channels of energy.

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arXiv2017

Soliton resolution along a sequence of times for the focusing energy critical wave equation

In this paper, we prove the soliton resolution conjecture for general type II solutions to the focusing energy critical wave equation, in space dimension 3,4 or 5, along a sequence of times. This is an important step towards the full soliton resolution in the nonradial case and without any size restrictions. This paper is an extension of the arXiv preprint 1510:00075 by the second author, where the finite time blow-up case is treated, with an error converging to zero in a weaker sense.

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arXiv2016

Universality of blow up profile for small blow up solutions to the energy critical wave map equation

In this paper we introduce the channel of energy argument to the study of energy critical wave maps into the sphere. More precisely, we prove a channel of energy type inequality for small energy wave maps, and as an application we show that for a wave map that has energy just above the degree one harmonic maps and that blows up in finite time, the solution asymptotically de-couples into a regular part plus a traveling wave with small momentum, in the energy space. In particular, the only possible form of energy concentration is through the concentration of traveling waves. This is often called quantization of energy at blow up. We also give a brief review of important background results in the subcritical and critical regularity theory for the two dimensional wave maps.

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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 $$\partial_tu=Δu+|u|^{\frac{4}{d-2}}u, \ \ x\in \mathbb R^d$$ 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

Profiles of bounded radial solutions of the focusing, energy-critical wave equation

In this paper we consider global and non-global bounded radial solutions of the focusing energy-critical wave equation in space dimension 3. We show that any of these solutions decouples, along a sequence of times that goes to the maximal time of existence, as a sum of modulated stationary solutions, a free radiation term and a term going to 0 in the energy space. In the case where there is only one stationary profile, we show that this expansion holds asymptotically without restriction to a subsequence.

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arXiv2015

Construction of multi-solitons for the energy-critical wave equation in dimension 5

We construct 2-solitons of any speed of the focusing energy-critical nonlinear wave equation in dimension 5. The existence result also holds for the case of K-solitons, for any K >2, assuming that the speeds are collinear. The main difficulty of the construction is the strong interaction between the solitons due to the slow spatial decay of the single soliton. This is in contrast with previous constructions of multi-solitons for other nonlinear models (like generalized KdV and nonlinear Schrodinger equations in energy subcritical cases), where the interactions are exponentially small in time due to the exponential decay of the solitons.

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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 Schrödinger 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

Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions

This is the first of two papers devoted to the study of the properties of the blow-up surface for the $N$ dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in one space dimension. Our goal here is to extend some of the properties to higher dimension. In dimension one, an essential tool was to study the dynamics of the solution in similarity variables, near the set of non-zero equilibria, which are obtained by a Lorentz transform of the space-independent solution. As a matter of fact, the main part of this paper is to study similar objects in higher dimensions. More precisely, near that set of equilibria, we show that solutions are either non-global, or go to zero, or converge to some explicit equilibrium. We also show that the first case cannot occur in the characteristic case, and that only the third possibility occurs in the non-characteristic case, thanks to the non-degeneracy of the blow-up limit, another new result in our paper. As a by-product of our techniques, we obtain the stability of the zero solution.

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arXiv2014

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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arXiv2014

On the nonexistence of pure multi-solitons for the quartic gKdV equation

We consider the quartic (nonintegrable) (gKdV) equation. Let u(t) be an outgoing 2-soliton of the equation, i.e. a solution behaving exactly as the sum of two solitons (of speeds c1 and c2) for large positive time. In arXiv:0910.3204, for nearly equal solitons, the solution u(t) is computed up to some order of epsilon=1-c2/c1, everywhere in time and space. In particular, it is deduced that u(t) is not a multi-soliton for large negative time, proving the nonexistence of pure multi-soliton in this context. In the present paper, we prove the same result for an explicit range of speeds: 3/4 c1< c2< c1, by a different approach, which does not longer require a precise description of the solution. In fact, the nonexistence result holds for outgoing N-solitons, for any N>1, under an explicit assumption on the speeds, which is a natural generalization of the condition for N=2.

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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

Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations

Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.

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arXiv2014

On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations

We consider a blow-up solution for the semilinear wave equation in $N$ dimensions, with subconformal power nonlinearity. Introducing $\RR_0$ the set of non-characteristic points with the Lorentz transform of the space-independent solution as asymptotic profile, we show that $\RR_0$ is open and that the blow-up surface is of class $C^1$ on $\RR_0$. Then, we show the stability of $\RR_0$ with respect to initial data.

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