In this text, we shall give an outline of some recent results (see \ccite{bahourichemin2} \ccite{bahourichemin3} and \ccite{bahourichemin4}) of local wellposedness for two types of quasilinear wave equations for initial data less regular than what is required by the energy method. To go below the regularity prescribed by the classical theory of strictly hyperbolic equations, we have to use the particular properties of the wave equation. The result concerning the first kind of equations must be understood as a Strichartz estimate for wave operators whose coefficients are only Lipschitz while the result concerning the second type of equations is reduced to the proof of a bilinear estimate for the product of two solutions for wave operators whose coefficients are not very regular. The purpose of this talk is to emphasise the importance of ideas coming from microlocal analysis to prove such results.
We use radial estimates for pseudodifferential operators to describe long time evolution of solutions to $ i u_t - P u = f $ where $ P $ is a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions and where $ f $ is smooth. This is motivated by recent results of Colin de Verdière and Saint-Raymond [arXiv:1801.05582] concerning a microlocal model of internal waves in stratified fluids.
We give a factorization procedure for a strictly hyperbolic partial differential operator of second order with logarithmic slow scale coefficients. From this we can microlocally diagonalize the full wave operator which results in a coupled system of two first-order pseudodifferential equations in a microlocal sense. Under the assumption that the full wave equation is microlocal regular in a fixed domain of the phase space, we can approximate the problem by two one-way wave equations where a dissipative term is added to suppress singularities outside the given domain. We obtain well-posedness o…
We introduce a global wave front set suitable for the analysis of tempered ultradistributions of quasianalytic Gelfand-Shilov type. We study the transformation properties of the wave front set and use them to give microlocal existence results for pull-backs and products. We further study quasianalytic microlocality for classes of localization and ultradifferential operators, and prove microellipticity for differential operators with polynomial coefficients.
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）
We discuss the microlocal Gevrey smoothing effect for the Schrödinger equation with variable coefficients via the propagation property of the wave front set of homogenous type. We apply the microlocal exponential estimates in a Gevrey case to prove our result.
We study an anisotropic version of the Shubin calculus of pseudodifferential operators on $\mathbf R^d$. Anisotropic symbols and Gabor wave front sets are defined in terms of decay or growth along curves in phase space of power type parametrized by one positive parameter that distinguishes space and frequency variables. We show that this gives subcalculi of Shubin's isotropic calculus, and we show a microlocal as well as a microelliptic inclusion in the framework. Finally we prove an inclusion for the anisotropic Gabor wave front set of chirp type oscillatory functions with a real polynomial p…
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic re…
Via minimization arguments and Concentration Compactness Principle, we prove the orbital stability of standing wave solutions for a class of quasilinear Schrödinger equation arising from physics.
We establish global existence in 3+1 dimensions of small-amplitude solutions of quasilinear Dirichlet-wave equations satisfying the null condition outside of star-shapped obstacles.
We prove the persistence of analyticity for classical solution of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy-Kowalewski and Ovsiannikov-Nirenberg theorems, lasts till a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of space-variables, then a classical solution also is analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space-analyticity (and the partial space-anal…
We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave equation by modifying Hörmander's method. Next, we construct an approximate solution, by solving our new reduced system given some scattering data at infinite time. Finally, we prove that the quasilinear wave equation has a global solution which agrees with the approximate solution at infinite time.
We consider the two-dimensional quasilinear wave equations with standard null-form type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.
We study the microlocal properties of bisingular operators, a class of operators on the product of two compact manifolds. We define a wave front set for such operators, and analyse its properties. We compare our wave front set with the $SG$ wave front set, a global wave front set which shares with it formal similarities.
We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation. Thus, we can handle all non-trapping obstacles. We are also able to handle non-diagonal systems satisfying the appropriate null condition.
In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in $H^{3}\times H^{2}$. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.
We prove the existence of approximate solutions in the (regular) Denjoy-Carleman sense for some systems of smooth complex vector fields. Such approximate solutions provide a well defined notion of Denjoy-Carleman wave front set of distributions on maximally real submanifolds in complex space which can be characterized in terms of the decay of the Fourier-Bros-Iagolnitzer transform. We also apply the approximate solutions to analyze the Denjoy-Carleman microlocal regularity of solutions of certain systems of first-order nonlinear partial differential equations.
In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the Laplacian of the compact base in terms of its eigenfunctions. For those terms corresponding to zero modes, we obtain decay using analogs of estimates of Klainerman and Sideris. For the nonzero modes, estimates for Klein-Gordon equations, which provide better decay, are available.
A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for $(1+1)$-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.
We extend the semilinear framework developed by the two authors and the non-trapping quasilinear theory developed by the first author to solve quasilinear wave equations with normally hyperbolic trapping. The most well-known example that fits into our general framework is wave-type equations on Kerr-de Sitter space. The key advance is an adaptation of the Nash-Moser iteration to our framework.
In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time $L^2$ estimates.
We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebra $\Gc(\Om)$ providing a way to measure the $\G$ and the $\Ginf$- regularity in $\LL(\Gc(\Om),\wt{\C})$. For the smaller family of functionals having a ``basic structure'' we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristic $\G$ and $\Ginf$-regularity.
We study long time existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. In particular, we obtain exterior domain analogs of the four dimensional results of Hörmander where the nonlinearity is permitted to depend on the solution not just its first and second derivatives. Previous proofs in exterior domains omitted this dependence as it did not mesh well with the energy methods in use.
In this article we study the quasilinear wave equation $\Box_{g(u, t, x)} u = 0$ where the metric $g(u, t, x)$ is close to the Schwarzschild metric. Under suitable assumptions of the metric coefficients, and assuming that the initial data for $u$ is small enough, we prove global existence of the solution. The main technical result of the paper is a local energy estimate for the linear wave equation on metrics with slow decay to the Schwarzschild metric.
Following the previous work [1], we investigate the impact of damping on the oscillation of smooth solutions to some kind of quasilinear wave equations with Robin and Dirichlet boundary condition. By using generalized Riccati transformation and technical inequality method, we give some sufficient conditions to guarantee the oscillation of all smooth solutions. From the results, we conclude that positive damping can ``hold back" oscillation. At last, some examples are presented to confirm our main results.
We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with \textbf{time dependent} inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely $C^1$ close to the Minkowski metric inside some large cylinder $\{\left.(t, x)\right||x|\leq R\}$ and approaches the Minkowski metric weakly as $|x|\rightarrow \infty$. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equati…
We study the boundary exact controllability for the quasilinear wave equation in the higher-dimensional case. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical condition. The Dirichlet action and the Neumann action …
We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations in which the classical non-local Eringen constitutive equation is generalized by employing space-fractional derivatives. Numerical examples illustrate the shape of solutions in dependence of the order of the space-fractional derivative.
For the quasilinear wave equation \partial_t^2u - Δu = u_t u_{tt}, we analyze the long-time behavior of classical solutions with small (not rotationally invariant) data. We give a complete asymptotic expansion of the lifespan and describe the solution close to the blowup point. It turns out that this solution is a ``blowup solution of cusp type,'' according to the terminology of the author.
We explore the global existence of solutions to systems of quasilinear wave equations satisfying the null condition when the initial data are sufficiently small. We adapt an approach of Keel, Smith, and Sogge, which relies on integrated local energy estimates and a weighted Sobolev estimate that yields decay in $|x|$, by using the $r^p$-weighted local energy estimates of Dafermos and Rodnianski. One advantage of this approach is that all time-dependent vector fields can be avoided and the proof can be readily adapted to address wave equations exterior to star-shaped obstacles.
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. A…
We are interested in the three-dimensional quasilinear wave equations with null condition. Global existence and pointwise decay for this model have been proved in the celebrated works of Klainerman \cite{Klainerman86} and Christodoulou \cite{Christodoulou86} for small smooth initial data. In this work, we illustrate the precise pointwise asymptotic behavior of the solutions for initial data posed on a hyperboloid and show that the decay rate $v^{-1}u^{-1}$ is optimal for a generic set of initial data.
We study the initial-boundary value problem of quasilinear wave equations outside of a star-shaped obstacle in four space dimensions, in which the nonlinear term under consideration may explicitly depend on the unknown function itself. By some new $L^{\infty}_{t}L^{2}_{x}$ and weighted $L^{2}_{t,x}$ estimates for the unknown function itself, together with energy estimates and KSS estimates, for the quasilinear obstacle problem we obtain a lower bound of the lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$, which coincides with the sharp lower bound of lifespan estimate for the co…
We prove the existence of solitary wave solutions to the quasilinear Benney system $$iu_{t}+u_{xx}=a|u|^pu+uv,\quad v_t+f(v)_x=(|u|^2)_x$$ where $f(v)=-γv^3$, $-1<p<+\infty$ and $a,γ>0$. We establish, in particular, the existence of travelling waves with speed arbitrary large if $p<0$ and arbitrary close to $0$ if $p>\frac 23$. We also show the existence of standing waves in the case $-1<p\leq \frac 23$, with compact support if $-1<p<0$.\\ Finally, we obtain, under certain conditions, the linearized stability of such solutions.
We consider quasilinear Schrödinger equations in $\mathbb{R}^{N}$ of the form% \[ -Δu+V(x)u-uΔ(u^{2})=g(u)\text{,}% \] where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schrödinger operator $-Δ+V$ is allowed to be indefinite, hence the variational functional does not satisfy the mountain pass geometry. By a local linking argument and Morse theory, we obtain a nontrivial solution for the problem. In case that $g$ is odd, we get an unbounded sequence of solutions.
The authors show that bilinear estimates for null forms hold for Dirichlet-wave equations outside of convex obstacle. This generalizes results for the Euclidean case of Klainerman and Machedon, and of Sogge for the variable coefficient boundaryless case. The estimates are used to prove a local existence theorem for semilinear wave equations satisfying the null condition.
We prove global existence of solutions to multiple speed, Dirichlet-wave equations with quadratic nonlinearities satisfying the null condition in the exterior of compact obstacles. This extends the result of our previous paper by allowing general higher order terms. In the currect setting, these terms are much more difficult to handle than for the free wave equation, and we do so using an analog of a pointwise estimate due to Kubota and Yokoyama.
We study global existence and decay estimates for quasilinear wave equations with dissipative terms in the Sobolev space $H^L \times H^{L-1}$, where $L \geq [d/2]+3$. The linear dissipative terms depend on space variable coefficient, and these terms may vanish in some compact region. For the proof of global existence, we need estimates of higher order energies. To control derivatives of the dissipative coefficient, we introduce an argument using the rescaling. Furthermore we get the decay estimates with additional assumptions on the initial data. To obtain the decay estimates, the rescaling ar…
We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function. Our method relies on a refined microlocal analysis linked to a second micr…
We study microlocal limits of plane waves on noncompact Riemannian manifolds (M,g) which are either Euclidean or asymptotically hyperbolic with curvature -1 near infinity. The plane waves E(z,ξ) are functions on M parametrized by the square root of energy z and the direction of the wave, ξ, interpreted as a point at infinity. If the trapped set K for the geodesic flow has Liouville measure zero, we show that, as z\to +\infty, E(z,ξ) microlocally converges to a measure μ_ξ, in average on energy intervals of fixed size, [z,z+1], and in ξ. We express the rate of convergence to the limit in terms …
In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu's seminal work [G. Luli, S. Yang, P. Yu, On one-dimension semi-linear wave equations with null conditions, Adv. Math.329 (2018) 174-188] from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense. In order to achieve these goals, we will employ Luli, Yang and Yu's weighted energy estimates with positive weights, intr…
We consider the global well-posedness of weak energy conservative solution to a general quasilinear wave equation through variational principle, where the solution may form finite time cusp singularity, when energy concentrates. As a main result in this paper, we construct a Finsler type optimal transport metric, then prove that the solution flow is Lipschitz under this metric. We also prove a generic regularity result by applying Thom's transversality theorem, then find piecewise smooth transportation paths among a dense set of solutions. The results in this paper are for large data solutions…
We consider a class of quasilinear wave equations in $3+1$ space-time dimensions that satisfy the "weak null condition" as defined by Lindblad and Rodnianski \cite{LR1}, and study the large time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is $-\partial_t^2 u + (1+u) Δu = 0$. Global solutions for such equations have been constructed by Lindblad \cite{Lindblad1,Lindblad2} and Alinhac \cite{Alinhac1}. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solution…
For the 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(\nabla u)\partial_{ij}u=0$ with coefficients independent of the solution $u$, a blowup result for small data solutions has been established in [1,2] provided that the null condition does not hold and a generic nondegeneracy condition on the initial data is fulfilled. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(u, \nabla u)\partial_{ij}u=0$ with coefficients that depend simultaneously on $u$ and $\nabla u$. When the null condition does not hold an…
