Sobolev Space H1 Research Articles

Well-posedness of the Cauchy problem for the fourth-order nonlinear Schrödinger equation

The sharp local well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation is established in the Sobolev space Hs(R) for s≥12, which improves the results in Huo and Jia (2007). In addition, we prove that this equation cannot be solved by an iteration scheme based on the Duhamel formula in Hs(R) for s<12. Our method relies upon the Bourgain space and a crucial bilinear estimate, which avoids the tedious classification of the location to the highest dispersion modulation.

Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on R. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.

Treatment of 3D diffusion problems with discontinuous coefficients and Dirac curvilinear sources

Three-dimensional diffusion problems with discontinuous coefficients and unidimensional Dirac sources arise in a number of fields. The statement we pursue is a singular-regular expansion where the singularity, capturing the stiff behavior of the potential, is expressed by a convolution formula using the Green kernel of the Laplace operator. The correction term, aimed at restoring the boundary conditions, fulfills a variational Poisson equation set in the Sobolev space H1, which can be approximated using finite element methods. The mathematical justification of the proposed expansion is the main focus, particularly when the variable diffusion coefficients are continuous, or have jumps. A computational study concludes the paper with some numerical examples. The potential is approximated by a combined method: (singularity, by integral formulas, correction, by linear finite elements). The convergence is discussed to highlight the practical benefits brought by different expansions, for continuous and discontinuous coefficients.

Needlets liberated

Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces Hs, but are easier to obtain numerically. With such rules we construct “generalised needlets”. The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper.

The existence and uniqueness of weak solutions for a highly nonlinear shallow-water model with Coriolis effect

In this paper, we consider the Cauchy problem for a highly nonlinear shallow water model arising from the full water waves with Coriolis effect. The existence of weak solutions to the equation in the lower order Sobolev space Hs(R) with 1&amp;lt;s≤32 is presented. Moreover, the local well-posedness of strong solutions in Sobolev space Hs(R) with s&amp;gt;32 is established by the pseudoparabolic regularization technique.

From Schrödinger equation to tempered distribution space on Minkowski chronotope and Schwartz Linear Algebra

In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.

The local well-posedness of the coupled Ostrovsky system with low regularity

In this paper, the Cauchy problem for the coupled Ostrovsky equations with an initial value in the Sobolev spaces Hs(R)×Hs(R) of lower order s is considered. With the bilinear estimate, it is proved that the initial value problem is locally well-posed in Hs(R)×Hs(R) for s>−34 by using Bourgain spaces. Moreover, if s<−34, it is demonstrated that one of the nonlinear iteration from the initial data to the putative solutions is discontinuous with an argument on the high-to-low frequency. In this sense, it is then concluded that the coupled Ostrovsky equations is ill-posed in Hs(R)×Hs(R) for s<−34.

On the vanishing viscosity limit for the viscous and resistive 3D magnetohydrodynamic system with axisymmetric data

The current paper aims to explore the three-dimensional axisymmetric viscous and resistive magnetohydrodynamic system. This article is concerned with two outcomes, in the first result we prove the global existence of a unique solution for this system under initial data lying in the Sobolev spaces Hs × Hs−2 with s&amp;gt;52, furthermore, we obtain uniform estimates with respect to the viscosity. The second outcome shows that the viscous and resistive solutions (vμ,bμ)μ&amp;gt;0 strongly converge to the non-viscous one (v, b). This convergence is performed in the space L∞(R+,Hs−2×Hs−2), also we obtain that the rate of convergence is (μt).

Dynamics of the black soliton in a regularized nonlinear Schrödinger equation

We consider a family of regularized defocusing nonlinear Schrödinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by a small perturbation in the Sobolev space H s ( R ) H^s(\mathbb {R}) with s &gt; 1 2 s &gt; \frac {1}{2} . We prove that the black soliton is spectrally stable (unstable) if the regularization parameter is below (above) some explicitly specified threshold. We illustrate the stable and unstable dynamics of the perturbed black solitons by using the numerical finite-difference method. The question of orbital stability of the black soliton is left open due to the mismatch of the function spaces for the energy and momentum conservation.

Positive solutions to the planar logarithmic Choquard equation with exponential nonlinearity

In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11].

The Continuous Wavelet Transform in Sobolev Spaces Over Locally Compact Abelian Group and Its Approximation Properties

The Sobolev space over locally compact abelian group Hs(G) is defined and we extend the continuous wavelet transform to Sobolev space Hs(G) for arbitrary real s. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces. Further, the asymptotic behaviour of the transforms of the L2 function for small scaling parameters is examined. In special cases, the wavelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.

Expected integration approximation under general equal measure partition

In this paper, we first use an L2−discrepancy bound to give the expected uniform integration approximation for functions in the Sobolev space H1(K) equipped with a reproducing kernel. The concept of stratified sampling under general equal measure partition is introduced into the research. For different sampling modes, we obtain a better convergence order O(N−1−1d) for the stratified sampling set than for the Monte Carlo sampling method and the Latin hypercube sampling method. Second, we give several expected uniform integration approximation bounds for functions equipped with boundary conditions in the general Sobolev space Fd,q∗, where 1p+1q=1. Probabilistic Lp−discrepancy bound under general equal measure partition, including the case of Hilbert space-filling curve-based sampling are employed. All of these give better general results than simple random sampling, and in particular, Hilbert space-filling curve-based sampling gives better results than simple random sampling for the appropriate sample size.

A solution operator for the \overline∂ equation in Sobolev spaces of negative index

Let Ω \Omega be a strictly pseudoconvex domain in C n \mathbb {C}^n with C k + 2 C^{k+2} boundary, k ≥ 1 k \geq 1 . We construct a ∂ ¯ \overline \partial solution operator (depending on k k ) that gains 1 2 \frac 12 derivative in the Sobolev space H s , p ( Ω ) H^{s,p} (\Omega ) for any 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty and s &gt; 1 p − k s&gt;\frac {1}{p} -k . If the domain is C ∞ C^{\infty } , then there exists a ∂ ¯ \overline \partial solution operator that gains 1 2 \frac 12 derivative in H s , p ( Ω ) H^{s,p}(\Omega ) for all s ∈ R s \in \mathbb {R} . We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of “anti-derivative operators” for distributions defined on bounded Lipschitz domains.

A note on the quartic generalized Korteweg–de Vries equation in weighted Sobolev spaces

In this paper we establish the persistence property for solutions of the quartic generalized Korteweg–de Vries equation with initial data in weighted Sobolev spaces Hs(R)∩L2(|x|2rdx) for s=1/12+ɛ and any r∈(0,R), for some 0<ɛ<1/4 and 0<R<s/2.

On the long-time asymptotic for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data

From the matrix ∂̄ problem, we propose a novel integrable coupled hierarchy with the application of recursive operator Λn which includes a coupled generalized nonlinear Schrödinger (cgNLS) equations with its Lax pair as n=0. We successfully derive infinite conservation laws and Hamiltonian function of the cgNLS equations for the first time. Furthermore, we employ the ∂̄ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space H1,1(R)=H1(R)∩L2,1(R). In a fixed space–time cone S(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt,x0∈[x1,x2],v∈[v1,v2]}, the long-time asymptotic behavior of the solutions u(x,t) and v(x,t) is derived. Based on the results of asymptotic behavior, we prove the soliton resolution conjecture of the cgNLS equations which consist of three terms, the soliton term confirmed by |Z(I)|-soliton on discrete spectrum, the t−12 order term on continuous spectrum and the residual error up to O(t−34).

Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric Sobolev space H1,p(X,d,m) associated with a positive and finite Borel measure m in a separable and complete metric space (X,d).We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space H1,2(P2(M),W2,m) arising from a positive and finite Borel measure m on the Kantorovich-Rubinstein-Wasserstein space (P2(M),W2) of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space M. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure m so that the resulting Cheeger energy is a Dirichlet form.We will eventually provide an explicit characterization for the corresponding notion of m-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the Γ-calculus inherited from the Dirichlet form.

On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system

We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic nonlinearity), we convert the Cauchy problem into a system of ordinary differential equations in a Banach space. Using this approach, we are able to demonstrate local well-posedness in the Sobolev space Hs where s>5/2. We also establish the continuity properties for the data-to-solution map for a range of Sobolev spaces. Finally, we briefly explore the relationship between the two-component system and the bi-Hamiltonian AKNS hierarchy.

The Majda-Biello system on the half-line

The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter α as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for 0<α<1 or 1<α<4 all these problems are well-posed for initial data in Sobolev spaces Hs, s≥0. For α=1 or α>4 well-posedness holds for Dirichlet data if s>−3/4, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For α=4 well-posedness of all problems holds for s≥3/4. The Robin and Neumann boundary data are in Hs/3 while the Dirichlet boundary data are in H(s+1)/3. These are consistent with the time regularity of the Cauchy problem for the corresponding linear system. The proof is based on linear estimates in Bourgain spaces derived by utilizing the Fokas solution formula for the forced linear system, and appropriate bilinear estimates suggested by the coupled nonlinearities. These show that the iteration map defined via the Fokas formula is a contraction in appropriate solution spaces. All the well-posedness results obtained her e are optimal.

On an improved PDE-based elliptic parameterization method for isogeometric analysis using preconditioned Anderson acceleration

Constructing an analysis-suitable parameterization for the computational domain from its boundary representation plays a crucial role in the isogeometric design-through-analysis pipeline. PDE-based elliptic grid generation is an effective method for generating high-quality parameterizations with rapid convergence properties for the planar case. However, it may generate non-uniform grid lines, especially near the concave/convex parts of the boundary. In the present work, we introduce a novel scaled discretization of harmonic mappings in the Sobolev space H1 to remit it. Analytical Jacobian matrices for the involved nonlinear equations are derived to accelerate the computation. To enhance the numerical stability and the speed of convergence, we propose a simple and yet effective preconditioned Anderson acceleration framework instead of using computationally expensive Newton-type iteration. Three preconditioning strategies are suggested, namely diagonal Jacobian, block-diagonal Jacobian, and full Jacobian. Furthermore, we discuss a delayed update strategy of the preconditioner, i.e., the preconditioner is updated every few steps to reduce the computational cost per iteration. Numerical experiments demonstrate the effectiveness and efficiency of our improved parameterization approach and the computational efficiency of our preconditioned Anderson acceleration scheme.

On local well-posedness of nonlinear dispersive equations with partially regular data

We revisit the local well-posedness theory of nonlinear Schrödinger and wave equations in Sobolev spaces Hs and H˙s, 0<s≤1. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.