This paper studies the local stable and unstable manifolds of equilibria for quasilinear and fully nonlinear PDEs. These manifolds are fundamental objects in the analysis of local dynamics. While their existence is well understood for ODEs, semilinear PDEs, and certain parabolic-type quasilinear PDEs, invariant manifold theorems are often unavailable for quasilinear PDEs whose nonlinearities involve a loss of regularity and whose linear parts do not provide sufficient smoothing.
Our main results establish the existence, uniqueness, and smoothness of local stable and unstable manifolds for nonlinear PDEs that satisfy suitable energy estimates. With the main focus on irrotational water waves with surface tension, this framework applies to a broad class of PDEs, including nonlinear Schrödinger equations, nonlinear wave equations, and the MMT model, as well as to certain gradient-type PDEs.
We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable blow-up criterion. The key idea is the use of the paradifferential operator calculus in order to reduce the fully nonlinear problem into an abstract quasilinear (stochastic) parabolic equation. This allows us to use some recent results on abstract quasilinear (stochastic) evolution equations in Banach spaces. To do so, we analyse the properties of the paradi…
This paper deals with the analysis of the internal control with constraint of positive kind of a parabolic PDE with nonlinear diffusion when the time horizon is large enough. The minimal controllability time will be strictly positive. We prove a global steady state constrained controllability result for a quasilinear parabolic with nonlinearity in the diffusion term. Then, under suitable dissipative assumption in the system and local controllability results, we conclude the result to any initial datum and any target trajectory.
We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, \begin{equation*} \operatorname{div}(A_s(x,u)\nabla u)=\operatorname{div}{\vec f}(x), \end{equation*} where $A_s(x,u)$ has "broken" derivatives of order $s\geq 0$, such as \begin{equation*} A_s(x,u) = a(x) + b(x)(u^+)^s, \end{equation*} with $(u^+)^0$ being understood as the characteristic function on $\{u>0\}$. The vector ${\vec f}(x)$ is assumed to be $C^α$ in case $s=0$, and $C^{1,α}$ (or higher) in case $s>0$. Using geometric methods, we prove almost complete r…
We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial-boundary…
We prove existence and uniqueness of global-in-time solutions in the $W^{-1,p}_D$-$W^{1,p}_D$-setting for abstract quasilinear parabolic PDEs with nonsmooth data and mixed boundary conditions, including a nonlinear source term with at most linear growth. Subsequently, we use a bootstrapping argument to achieve improved regularity of these global-in-time solutions within the functional-analytic setting of the interpolation scale of Bessel-potential dual spaces $H^{θ-1,p}_D = [W^{-1,p}_D,L^p]_θ$ with $θ\in [0,1]$ for the abstract equation under suitable additional assumptions. This is done by me…
We investigate quasilinear discrete PDEs $\partial_t u = Δ^N φ(u)+ Kf(u)$ of reaction-diffusion type with nonlinear diffusion term defined on an $n$-dimensional unit torus discretized with mesh size $\tfrac1N$ for $N\in {\mathbb N}$, where $Δ^N$ is the discrete Laplacian, $φ$ is a strictly increasing $C^5$ function and $f$ is a $C^1$ function. We establish $L^\infty$ bounds and space-time Hölder estimates, both uniform in $N$, of the first and second spatial discrete derivatives of the solutions. In the equation, $K>0$ is a large constant and we show how these estimates depend on $K$. The moti…
We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli' study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we i…
Existence and uniqueness of weak solutions to quasilinear PDEs with critical data
Université Paris-Saclay（巴黎萨克雷大学）；France-Australia Mathematical Sciences and Interactions ANU - CNRS International Research Laboratory（法澳数学科学互动ANU-CNRS国际研究实验室）；CNRS Laboratoire de Mathématiques d’Orsay（法国国家科学研究中心奥赛数学实验室）
In this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension two (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time-resonance structure and modified scattering. It is slightly simpler, however, at the technical level and our goal here is to present our method in this simplified situation. The full system is the subject of a forthcoming paper.
In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the presence of the Carleman Weight Function. Compared with previous publications, the main novelty of this paper is that the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed. The method works for both ill-posed Cauchy problems for some quasilinear PDEs of the second order and for some Coefficient Inverse Problems. However, to si…
We consider in this article the system of gravity-capillary waves in all dimensions and under the Zakharov/Craig-Sulem formulation. Using a paradifferential approach introduced by Alazard-Burq-Zuily, we symmetrize this system into a quasilinear dispersive equation whose principal part is of order $3/2$. The main novelty, compared to earlier studies, is that this reduction is performed at the Sobolev regularity of quasilinear pdes: $H^s(R^d)$ with $s\textgreater{}3/2+d/2$, $d$ being the dimension of the free surface. From this reduction, we deduce a blow-up criterion involving solely the Lipsch…
Some higher-order quasilinear parabolic, hyperbolic, and nonlinear dispersion equations are shown to admit various blow-up, extinction, and travelling wave solutions, which reduce to variational problems admitting countable families of compactly supported solutions.
Long time strong Sobolev instability for quasilinear NLS
University of Vienna（维也纳大学）；Politecnico di Milano（米兰理工大学）；Università della Calabria（卡拉布里亚大学）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）
In this work, we consider the problem of boundary stabilization for a quasilinear 2X2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H^2 exponential stability of the closedloop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4X4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and success…
Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipativ…
The paper deals with the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the question of the nonlinear interaction of a plane wave with its reflection off a vertical wall. The main result is the construction of small amplitude, standing (namely periodic in time and space, and not travelling) solutions of Sobolev regularity, for almost all values of the surface tension coefficient, and for a large set of time-frequencies. This is an existence result for a quasi-linear, Hamiltonian, reversible system of two autonomous pseudo-PDEs with small divisors. The p…
We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.
We give a construction allowing to construct local renormalised solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalising the recent results of [BDH16,FG16,OW16]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [Hai14,BHZ16,CH16] applies, including in particular one-dimensional systems with KPZ-type nonlinearities driven by space-time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renor…
We study existence and regularity of weak solutions for the following PDE $$ -\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in $B_1$}. $$ where $A(x,s) = A_+(x)χ_{\{s>0\}}+A_-(x)χ_{\{s\le 0\}}$ and $f(x,s) = f_+(x)χ_{\{s>0\}}+f_-(x)χ_{\{s\le 0\}}$. Under the ellipticity assumption that $\frac{1}μ\le A_{\pm} \le μ$, $A_{\pm}\in C(Ø)$ and $f_{\pm}\in L^N(Ø)$, we prove that under appropriate conditions the PDE above admits a weak solution in $W^{1,p}(B_1)$ which is also $C^{0,α}_{loc}$ for every $α\in (0,1)$ with precise estimates. Our methods relies on similar techniques as those deve…
Periodic operators over a component domain and homogenization of some class of quasi-linear elliptic problems in two-component domain with interfacial resistance
We prove an almost global in time existence result of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that usual paradifferential calculus…
We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of $\varepsilon^{-5/3+}$, for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of…
The motion of both internal and surface waves in incompressible fluids under capillary and gravity forces is a major research topic. In particular, we review the derivation of some new models describing the dynamics of gravity-capillary nonlinear waves in incompressible flows. These models take the form of both bidirectional and unidirectional nonlinear and nonlocal wave equations. More precisely, with the goal of telling a more complete story, in this paper we present the results in the works \cite{GS,aurther2019rigorous,granero2019models,granero2019well,granero2020well,granero2021motion, gra…
The stability of periodic traveling wave solutions to dispersive PDEs with respect to `arbitrary' perturbations is still widely open. The focus is put here on stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg--de Vries equation (qKdV), and the Euler--Korteweg sys…
Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature of the Hessian matrix of action integrals associated with those waves. In a previous paper (Nonlinearity 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg--de Vries equation and dispersive perturbations of the Euler equation…
We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - σ(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $ C^{α-2}$ for $α> \frac{2}{3}$ on the parabolic Hölder scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, …
Weak solutions for weak turbulence models in electrostatic plasmas
Institute for Fusion Studies, University of Texas at Austin（德克萨斯大学奥斯汀分校聚变研究所）；Oden Institute for Computational Sciences and Engineering, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算科学与工程研究所）
Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation
Université Paris Dauphine（巴黎第九大学）；Humboldt-Universitaet zu Berlin（柏林洪堡大学）；Institut für Mathematik（数学研究所）；The University of Edinburgh（爱丁堡大学）；Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）；Institut Universitaire de France（法国高等研究院）；Hausdorff Center for Mathematics（豪斯多夫数学中心）；Institut für Angewandte Mathematik Universität Bonn（波恩大学应用数学研究所）；University of Oxford（牛津大学）；Beijing Institute of Technology（北京理工大学）
We prove the existence of time-periodic, small amplitude solutions of autonomous quasilinear or fully nonlinear completely resonant pseudo-PDEs of Benjamin-Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero. At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions). The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation tha…
