arXiv subjects
Publications and source records attributed to Frank Merle.
We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.
We consider the radial energy-critical non-linear focusing Schrödinger equation in dimension N=3,4,5. An explicit stationnary solution, W, of this equation is known. In a previous work by C. Carlos and F. Merle, the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u)=E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W.
We consider the energy-critical non-linear focusing wave equation in dimension N=3,4,5. An explicit stationnary solution, $W$, of this equation is known. The energy E(W,0) has been shown by C. Kenig and F. Merle to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u_0,u_1)=E(W,0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analoguous to our previous work on radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.
We consider the generalized Korteweg-de Vries equation \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}, (1) with general $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of (1) of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. In this paper, under general assumptions on $f$ and $Q_c$, we prove that the family of soliton solutions around $Q_c$ is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\geq 0$), then $u(t)$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$. Note in particular that we do not assume the stability of $Q_{c}$. This result is based on a rigidity property of equation (1) around $Q_{c}$ in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in previous works devoted to the pure power case.
We consider the generalized Korteweg-de Vries equation $$ \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}$$ with general $C^2$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. Stability theory for $Q_c$ is well-known. In previous works, we have proved that for $f(u)=u^p$, $p=2,3,4$, the family of solitons is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\geq 0$), then $u(t,.+ρ(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+\sim c$. Then, the asymptotic stability result could be extended to the case of general assumptions on $f$ and $Q_c$. The objective of this paper is twofold. The main objective is to prove that in the case $f(u)=u^p$, $p=2,3,4$, $ρ(t)-c_+ t$ has limit as $t\to +\infty$ under the additional assumption $x_+ u\in L^2$. The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity $f(u)$, when the ratio of the speeds of the solitons is small. The motivation is to accompany forthcoming works devoted to the collision of two solitons in the nonintegrable case. The arguments are refinements of previous works specialized to the case $u(t)\sim Q_{c_1}+Q_{c_2}$, for $0< c_2 \ll c_1$.
We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with data of energy smaller than or equal to a harmonic map of minimal energy. An alternative holds: either the data is the harmonic map and the soltuion is constant in time, or the solution scatters in infinite time.
This paper concerns the problem of collision of two solitons for the quartic generalized Korteweg-de Vries equation. We introduce a new framework to describe the collision in the special case where one soliton is small with respect to the other. We prove that the two soliton survive the collision, we describe the collision phenomenon (computation of the first order of the resulting shifts on the solitons). Moreover, we prove that in this situation, there does not exist pure two-soliton solutions.
We continue our study of the collision of two solitons for the subcritical generalized KdV equations. In a previous paper, mainly devoted to the case of the quartic gKdV equation, we have introduced a new framework to understand the collision of two solitons in the case where one soliton is small with respect to the other. In this paper, we consider the case of a general nonlinearity $f(u)$ for which the two solitons are nonlinearly stable. We prove that in this situation the two solitons survive and we describe the collision at the main orders.
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 $t\to T$.
We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy holds: a) if the homogeneous Sobolev norm H^1 of u_0 is smaller than that of W, we have global well-posedness and scattering, b) if the homogeneous Sobolev norm H^1 of u_0 is larger than that of W, there is blow-up in finite time. Our general approach is the one we introduced in our previous work on the corresponding problem for NLS (math.AP/0610266, Inventiones Math 2006, Online First), where we proved the corresponding result for NLS in the radial case. In the case of the wave equation we are able to treat general data by using a further conservation law in the energy space, the finite speed of propagation and Lorentz transformations to establish a crucial orthogonality property for " energy critical " elements. To prove the required rigidity theorem in the case of blow-up in finite time, we cannot use the invariance of the L^2 norm as in the case of NLS. Instead, (following earlier work of Merle-Zaag and of Giga-Kohn in the parabolic case) we introduce self-similar variables. We thus find a further Liapunov function which allows us to reduce matters to a degenerate elliptic problem with critical non-linearity. We use unique continuation to rule out the existence of non-zero solutions for the degenerate elliptic problem, thus completing the proof.
We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.
The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H^1(L^2 norm and energy). We consider in this paper the {\it critical} generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H^1 uniform in time for all H^1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)|_{H^1} \to +\infty as T\uparrow T, where T\le +\infty (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L^2 mass close to the minimal mass allowing blow up and with decay in L^2 at the right, we prove after rescaling and translation which leave invariant the L^2 norm that the solution converges to a {\it universal} profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.
We prove in this paper the stability and asymptotic stability in H^1 of a decoupled sum of N solitons for the subcritical generalized KdV equations $u_t+(u_{xx}+u^p)_x=0$ (1<p<5). The proof of the stability result is based on energy arguments and monotonicity of local L^2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [15].