Low Regularity Spaces Research Articles

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.

Well-posedness of the Cauchy problem for the kinetic DNLS on T

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus [Formula: see text] for [Formula: see text], where the constants [Formula: see text] are such that [Formula: see text] and [Formula: see text], and [Formula: see text] denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces [Formula: see text] for [Formula: see text]. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when [Formula: see text], cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in [Formula: see text], [Formula: see text]. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term [Formula: see text], in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.

Low regularity well–posedness for dispersive equations on semi–infinite intervals

We summarize and report recent advancements in the theory of dispersive equations posed on semi–infinite intervals. By employing modern Fourier analysis techniques we show how these initial-boundary value problems can be resolved for initial data in Sobolev spaces of low regularity. In almost all the cases, we obtain sharp results that match the regularity of their full line analogues. We especially address the issues of local well–posedness, nonlinear smoothing, and uniqueness of solutions. In the second part of the paper we apply some of the tools we discussed in the first part, to prove existence and uniqueness of $ L^2 $ solutions for the biharmonic equation, extending the result in [10].

Convergence problem of Schrödinger equation and wave equation in low regularity spaces

This paper is devoted to investigating the convergence problem of the nonlinear Schrödinger equation and the nonlinear wave equation. Firstly, the almost everywhere pointwise convergence of the one-dimensional nonlinear Schrödinger equation is established in Hs(R)(s≥14) with the aid of the decomposition of integral form solution to problem. Secondly, the uniform convergencelimt→0⁡supx∈R⁡|u(x,t)−S1(t)f|=0 is established with initial data in Hs(R)(s>0), where u is the solution to the one-dimensional nonlinear Schrödinger equation and S1(t)f is the solution to the free one-dimensional Schrödinger equation. Finally, the almost everywhere pointwise convergence of the nonlinear wave equation is established in Hs(R3)(s>1).

Symplectic nonsqueezing for the KdV flow on the line

We show symplectic non-squeezing for the KdV equation on the line $\mathbb R$. This is achieved via finite-dimensional approximation. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough of Killip and Visan in the well-posedness theory of KdV in low regularity spaces, relying on its completely integrable structure. We also prove and exploit a weak well-posedness result for KdV in $H^{-1}(\mathbb R)$. Furthermore, the employment of our methods provides a new concise proof for the known result of symplectic non-squeezing for the same equation on the circle $\mathbb T$, first proved by Colliander, Keel, Staffilani, Takaoka, and Tao.

Long-time asymptotic behavior of the coupled dispersive AB system in low regularity spaces

In this paper, we mainly investigate the long-time asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann–Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the long-time asymptotic behavior of these solutions. We demonstrate that in any fixed time cone Cx1,x2,v1,v2=(x,t)∈R2∣x=x0+vt,x0∈x1,x2,v∈v1,v2, the long-time asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by N(I) solitons on the discrete spectrum, the leading order term O(t−1/2) on the continuous spectrum, and the allowable residual O(t−3/4).

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner‐type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.

On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity

We study the nonlinear Schrödinger equation with initial data in $ \mathcal Z ^s_p(\mathbb R^d)=\dot{H}^s(\mathbb R^d)\cap L^p(\mathbb R^d)$, where $0 < s < \min\{d/2,1\}$ and $2 < p < 2d/(d-2s)$. After showing that the linear Schrödinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.

The Korteweg–de Vries equation on the half-line with Robin and Neumann data in low regularity spaces

The well-posedness of the initial–boundary value problem (ibvp) for the Korteweg–de Vries equation on the half-line is studied for initial data u0(x) in spatial Sobolev spaces Hs(0,∞), s>−3/4, and Robin and Neumann boundary data φ(t) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear equation. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear equation. Then, using these and the needed bilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.

Parameter space study of optimal scale-dependent weights in TV image denoising

We investigate the choice of finite-dimensional parameter spaces within a bilevel optimization framework for selecting scale-dependent weights in total variation image denoising with non-uniform noise. Due to the pointwise box constraints on the parameter function, we prove existence of Lagrange multipliers in low regularity spaces, and derive a first-order optimality system. To cope with the difficulties related to the lack of regularity, a Moreau-Yosida regularization is introduced and convergence of the regularized optimal parameters towards the optimal weight for the original problem is verified. For each regularized bilevel problem a second-order quasi-Newton algorithm is proposed, together with a semismooth Newton scheme for solving the lower-level problem. Finally, several numerical tests are carried out to compare the different parameter space choices and draw some conclusions.

Local Well-Posedness of Skew Mean Curvature Flow for\xa0Small Data in dge 4 Dimensions

The skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in {{mathbb {R}}}^{d+2} (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension dge 4.

Elliptic problems in Besov and Sobolev—Triebel—Lizorkin spaces of low regularity

Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev–Triebel–Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.

Long‐time asymptotic behavior of the fifth‐order modified KdV equation in low regularity spaces

AbstractBased on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems and the Dbar approach, the long‐time asymptotic behavior of solutions to the fifth‐order modified KdV (Korteweg–de Vries) equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of the ‐method, we give its global well‐posedness in lower regularity Sobolev spaces and then obtain the asymptotic behavior in these spaces with weights.

Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q &gt; d q&gt;d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.

Soliton resolution for the focusing modified KdV equation

The soliton resolution for the focusing modified Korteweg-de Vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through ∂‾-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory x=vt for any fixed v. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As by-products of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.

Global attractor for weakly damped, forced mKdV equation in low regularity spaces

The modified Korteweg-de Vries equation (for short, mKdV) models the propagation of nonlinear water waves in the shallow water approximation. We consider the weakly damped and forced mKdV under the periodic boundary condition. We often study mKdV equation by using the Miura transformation, which converts solutions of mKdV to solutions of the Korteweg-de Vries equation (KdV). However, if mKdV has damping and external forcing terms, the Miura transformation does not work well. To see the asymptotic behaviour of the solutions of mKdV equation, the study of global attractor is important. We prove the existence of the global attractor for $$ {s > 9/10} $$ in $$\dot{H}^{s}$$ for mKdV equation by applying the modified energies and almost conserved quantities.

The IVP for the evolution equation of wave fronts in chemical reactions in low-regularity Sobolev spaces

In this work, we study the initial-value problem for an equation of evolution of wave fronts in chemical reactions. We show that the associated initial-value problem is locally and globally well-posed in Sobolev spaces $$H^s({\mathbb {R}})$$ , where $$s>1/2$$ . The well-posedness in critical space $$\dot{H}^{1/2}(\mathbb R)$$ for small initial data is obtained. We also show that our result is sharp, in the sense that the flow-map data solution is not $$C^2$$ at origin, for $$s<1/2$$ . Furthermore, we study the behavior of the solutions when $$\mu \downarrow 0$$ .

Continuous Dependence of Solutions in Low Regularity Spaces for the Hall-MHD Equations

In this paper, we consider the strong solutions to the incompressible viscous, resistive Hall-MHD system. Firstly, we show that the solutions to the Hall-MHD system depend continuously on the initial data in $$H^s({\mathbb {R}}^3)$$ , $$s>\frac{3}{2}$$ . In addition, we obtain that the solutions to the Hall-MHD system converge to the solutions of the MHD system as the Hall effect coefficient tends to zero.

Uniform stabilization of Boussinesq systems in critical &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbf{L}^q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension $(d-1)$ for the fluid component and of dimension $1$ for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to $\mathbf{L}^3(\Omega$) for $ d = 3 $) and the space $L^q(\Omega$) for the thermal component, $ q > d $. Thus, this paper may be viewed as an extension of \cite{LPT.1}, where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.

Short time existence for the curve diffusion flow with a contact angle

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle α∈(0,π): The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class W2γ with γ∈(32,2]. For the proof the evolving curve is represented by a height function over a reference curve: The local well-posedness of the resulting quasilinear, parabolic, fourth-order PDE for the height function is proven with the help of contraction mapping principle. Difficulties arise due to the low regularity of the initial curve. To this end, we have to establish suitable product estimates in time weighted anisotropic L2-Sobolev spaces of low regularity for proving that the non-linearities are well-defined and contractive for small times.