Local Smoothing Effect Research Articles

Sharp well-posedness for the cubic NLS and mKdV in

Abstract We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{\mathbb {R}}})$ for any regularity $s>-\frac 12$ . Well-posedness has long been known for $s\geq 0$ , see [55], but not previously for any $s<0$ . The scaling-critical value $s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in $H^s({{\mathbb {R}}})$ for any $s>-\frac 12$ . The best regularity achieved previously was $s\geq \tfrac 14$ (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.

An inhomogeneous porous medium equation with large data: Well-posedness

We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spatial infinity like |x|−γ, for γ∈[0,2), and is allowed to be singular at the origin. In particular we show local-in-time existence and uniqueness for a class of large initial data which includes as “endpoints” those growing at a rate of |x|(2−γ)/(m−1), in a weighted L1-average sense. We also identify global-existence and blow-up classes, whose respective forms strongly support the claim that such a growth rate is optimal, at least for positive solutions. As a crucial step in our existence proof we establish a local smoothing effect for large data without resorting to the classical Aronson-Bénilan inequality and using the Bénilan-Crandall inequality instead, which may be of independent interest since the latter holds in much more general settings.

Gains of integrability and local smoothing effects for quadratic evolution equations

We characterize geometrically the semigroups generated by non-selfadjoint quadratic differential operators (e−tqw)t≥0 enjoying local smoothing effects and providing gains of integrability. More precisely, we prove that the evolution operators e−tqw map Lp on Lq∩C∞, for all 1≤p≤q≤∞, if and only if the singular space of the quadratic operator qw is included in the graph of a linear map. We also provide quantitative estimates for the associated operator norms in the short-time asymptotics 0<t≪1.

On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrödinger Equation with Fractional Dissipation

In this paper, we are concerned with the nonlinear fractional Schrödinger equation. We extend the result of Guo and Huo and prove that the Cauchy problem of the nonlinear fractional Schrödinger equation is global well-posed in H32−γ(R) with 12≤γ&lt;1. In view of the complexity of the nonlinear fractional Schrödinger equation itself, the local smoothing effect and maximal function estimates are not enough for presenting the global well-posedness for the nonlinear fractional Schrödinger equation. In this paper, we use a suitably iterative scheme and complete the global well-posed result for Equation (R). Moreover, we obtain the orbital stability of standing waves for the above equations via establishing the profile decomposition of bounded sequences in Hs(RN) (0&lt;s&lt;1) with N≥2.

Local well-posedness and spatial analyticity for the generalized Zakharov–Kuznetsov equation

The initial value problem for the generalized Zakharov–Kuznetsov equation on R2 is shown to be local well-posed in spaces of functions which are analytic on a strip without shrinking the width of the strip in time. The proof mainly relies upon the local smoothing effect, a multi-dimensional maximal function estimate and Bourgain-type spaces. These techniques are significant for derivative nonlinear dispersive equation in low regularity spaces. In addition, under the boundedness assumption of a suitable Sobolev norm the generalized Zakharov–Kuznetsov equation is proved to be Gevrey-class analysis. Especially, we obtain an explicit lower bound on the possible decreasing rate of the uniform radius of analyticity of a solution starting from analytic initial data.

On Kato's smoothing effect for a fractional version of the Zakharov-Kuznetsov equation

In this work, we study some regularity properties associated with the initial value problem (IVP)$\begin{align} \left\{ \begin{array}{ll} \partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u = 0, \quad 0&lt; \alpha\leq 2, &amp; \\ u(x, 0) = u_{0}(x), \quad x = (x_{1}, x_{2}, \dots, x_{n})\in \mathbb{R}^{n}, \, n\geq 2, \quad t\in\mathbb{R}, &amp; \ \end{array} \right. \end{align}\ \ \ \ \ (1)$where $ (-\Delta)^{\alpha/2} $ denotes the $ n- $dimensional fractional Laplacian.We show that solutions to the IVP (1) with initial data in a suitable Sobolev space exhibit a local smoothing effect in the spatial variable of $ \frac{\alpha}{2} $ derivatives, almost everywhere in time. One of the main difficulties that emerge when trying to obtain this regularizing effect underlies that the operator in consideration is non-local, and the property we are trying to describe is local, so new ideas are required. Nevertheless, to avoid these problems, we use a perturbation argument replacing $ (-\Delta)^{\frac{\alpha}{2}} $ by $ (I-\Delta)^{\frac{\alpha}{2}}, $ that through the use of pseudo-differential calculus allows us to show that solutions become locally smoother by $ \frac{\alpha}{2} $ of a derivative in all spatial directions.As a by-product, we use this particular smoothing effect to show that the extra regularity of the initial data on some distinguished subsets of the Euclidean space is propagated by the flow solution with infinity speed.

Scattering for 3D Cubic Focusing NLS on the Domain Outside a Convex Obstacle Revisited

In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in ℝ3 with Dirichlet boundary conditions. We revisit the scattering result below ground state in Killip—Visan—Zhang [The focusing cubic NLS on exterior domains in three dimensions. Appl. Math. Res. Express. AMRX, 1, 146–180 (2016)] by utilizing the method of Dodson and Murphy [A new proof of scattering below the ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc., 145, 4859–4867 (2017)] and the dispersive estimate in Ivanovici and Lebeau [Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples. Comp. Rend. Math., 355, 774–779 (2017)], which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimates, we prove that the solution decays at a large time interval, which meets the scattering criterion.

Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

By [Formula: see text]-method, the interaction Morawetz estimate, long-time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation [Formula: see text] where [Formula: see text], and [Formula: see text], [Formula: see text], with radial data in [Formula: see text] for [Formula: see text]. It is a sharp global result except the critical case [Formula: see text], which is a very difficult open problem.

Pointwise Convergence of Solutions to the Schrödinger Equation on Manifolds

AbstractLet $(M^{n},g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^{\unicode[STIX]{x1D6FC}}(M)$. For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that $\unicode[STIX]{x1D6FC}&gt;\frac{1}{2}$ is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound $\unicode[STIX]{x1D6FC}&gt;1$ from interpolation. We managed to go below 1 for dimension ${\leqslant}$ 3. The more interesting thing is that, for a one-dimensional compact manifold, $\unicode[STIX]{x1D6FC}&gt;\frac{1}{3}$ is sufficient.

A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation

In this paper we prove the convergence of a Crank-Nicolson type Galerkin finite element scheme for the initial value problem associated to the Benjamin-Ono equation. The proof is based on a recent result for a similar discrete scheme for the Korteweg-de Vries equation and utilizes a local smoothing effect to bound the $H^{1/2}$-norm of the approximations locally. This enables us to show that the scheme converges strongly in $L^{2}(0,T;L^{2}_{\text{loc}}(\mathbb{R}))$ to a weak solution of the equation for initial data in $L^{2}(\mathbb{R})$ and some $T > 0$. Finally we illustrate the method with some numerical examples.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

Local energy decay and smoothing effect for the damped Schrödinger equation

We prove the local energy decay and the smoothing effect for the damped Schr{\"o}dinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation which we consider.

A Fourth-Order Dispersive Flow Equation for Closed Curves on Compact Riemann Surfaces

A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main result, time-local existence and uniqueness of a solution to the initial value problem are established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the absence of the local smoothing effect. The proof is based on the geometric energy method combined with a kind of gauge transformation to eliminate the loss of derivatives. Specifically, to show the uniqueness of the solution, the detailed geometric analysis of the solvable structure for the equation is presented.

Exponential decay for a locally damped fifth-order equation posed on the line

We prove the exponential decay of the energy related to a locally damped fifth-order equation posed on the whole real line with the initial datum from a bounded set of L2. A local smoothing effect in H2 is established, which is essential to obtain the necessary a priory estimates. Moreover, it is shown that arguments used in the article can be applied to prove the exponential decay rate of solutions for the Korteweg–de Vries equation with a similar localized damping term provided the initial data are uniformly bounded in L2. This last fact improves some previous results.

A Fourth-Order Dispersive Flow into Kähler Manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.

Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the “elastic” operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term. We also provide counterexamples in order to show the optimality of our results.

Exponential Stabilization for the Nonlinear Schrödinger Equation with Localized Damping

Results of exponential stability related to the nonlinear Schrodinger equation posed on the whole real line will be discussed in this paper. We shall use local smoothing effect associated with the semigroup of the Schrodinger equation, Strichartz estimates, and compactness tools to determine the exponential decay of the energy, by assuming the initial data in convenient bounded sets of L2.

Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains

The Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete, and noncompact n dimensional Riemannian manifold (ℳ, g) is considered. Assuming that (ℳ, g) is nontrapping and, in addition, that the damping term is effective in ℳ \\ Ω, where Ω ⊂ ⊂ ℳ is an open bounded and connected subset with smooth boundary ∂Ω, such that is a compact set, exponential and uniform decay rates of the L 2 − level energy are established. The main ingredients in the proof of the exponential stability are: (A) an unique continuation property for the linear problem; and (B) a local smoothing effect for the linear and nonhomogeneous associated problem.

Local smoothing effect on the Schrödinger equation with harmonic potential

We consider the linear Schrödinger equation with repulsive harmonic potential. We establish the local smoothing effect of this type of equations. Our work extends the related results obtained by L. Vega and N. Visciglia for the free Schrödinger equation.

Weak solutions to initial-boundary-value problems for quasilinear evolution equations of an odd order

Initial-boundary-value problems in three different domains are considered for quasilinear evolution partial differential equations of an odd (not less than third) order with respect to spatial variables in the multidimensional case. The nonlinearity has the divergent form and at most a quadratic rate of growth. Assumptions on the differential operator of odd order provide global estimates on solutions in $L_2$ and a local smoothing effect. Results on existence and uniqueness of global weak solutions are established. The essential part of the study is the construction of special solutions to the corresponding linear equations of the "boundary potential" type, which ensures the results under natural smoothness assumptions on initial and boundary data provided we have certain relations between the dimension and the order of the equations.