Sobolev Norms Research Articles

A characterization of the subspace of radially symmetric functions in Sobolev spaces

In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.

Weak Galerkin finite element methods for optimal control problems governed by second order elliptic equations

This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new scheme.

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

Families of functionals representing Sobolev norms

Families of functionals representing Sobolev norms

Image Reconstruction Requirements for Short-Range Inductive Sensors Used in Single-Coil MIT.

MIT (magnetic induction tomography) image reconstruction from data acquired with a single, small inductive sensor has unique requirements not found in other imaging modalities. During the course of scanning over a target, measured inductive loss decreases rapidly with distance from the target boundary. Since inductive loss exists even at infinite separation due to losses internal to the sensor, all other measurements made in the vicinity of the target require subtraction of the infinite-separation loss. This is accomplished naturally by treating infinite-separation loss as an unknown. Furthermore, since contributions to inductive loss decline with greater depth into a conductive target, regularization penalties must be decreased with depth. A pair of squared L2 penalty norms are combined to form a 2-term Sobolev norm, including a zero-order penalty that penalizes solution departures from a default solution and a first-order penalty that promotes smoothness. While constraining the solution to be non-negative and bounded from above, the algorithm is used to perform image reconstruction on scan data obtained over a 4.3 cm thick phantom consisting of bone-like features embedded in agarose gel, with the latter having a nominal conductivity of 1.4 S/m.

Compact Embeddings for Fractional Super and Sub Harmonic Functions with Radial Symmetry

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

Novel Numerical Investigation of Reaction Diffusion Equation Arising in Oil Price Modeling

Consideration is given to a reaction–diffusion free boundary value problem with one or two turning points arising in oil price modeling. First, an exact (analytical) solution to the reduced problem (i.e., no diffusion term) was obtained for some given parameters. The space–time Chebyshev pseudospectral and superconsistent Chebyshev collocation method is proposed for both reaction diffusion (RDFBP) and reduced free boundary value problem. Error bounds on the discrete L2–norm and Sobolev norm (Hp) are presented. Adaptively graded intervals were introduced and used according to the value of turning points to avoid the twin boundary layers phenomena. Excellent convergent (spectrally) and stable results for some special turning points were obtained for both reduced and RDFBP equations on an adaptively graded interval and this has been documented for the first time.

Optimal decay rates in Sobolev norms for singular values of integral operators

The regularity of integration kernels forces decay rates of singular values of associated integral operators. This is well-known for symmetric operators with kernels defined on (a,b)×(a,b), where (a,b) is an interval. Over time, many authors have studied this case in detail [9,10,4,15,12]. The case of spheres has also been resolved [3]. A few authors have examined the higher dimensional case or the case of manifolds [1,6]. Typically, these authors have provided decay estimates of singular values in lp norms, p≥1 or in case of faster decay due to regularity, lp quasi-norms, 0<p<1. With that approach, it is straightforward to show that their estimates are optimal using periodic kernels obtained from Fourier series. Our new approach for deriving decay estimates of these singular values uses Weyl's asymptotic formula for Neumann eigenvalues [21] that we combine to an appropriately defined inverse Laplacian. We obtain decay estimates in the form nα where for the n -th singular value where α depends on dimension and on the Sobolev regularity of the kernel. Since we are interested in optimal estimates in case of regular kernels, instead of writing an upper bound by a constant times nα, we use the singular values of the kernel obtained by differentiation. While in [1,6]lp estimates were proven to be optimal by simply considering periodic Fourier series, these Fourier series do not provide sharp results for our estimates. Instead, series of Neumann eigenfunctions for the Laplacian that are specific to the domain of interest are used to prove that our decay estimates are optimal. Finally, we cover the case of real analytic kernels where we are also able to derive optimal estimates.

On the growth of Sobolev norm for the cubic NLS on two dimensional product space

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocussing nonlinear Schrödinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for frequency localized functions, which estimates are used for enhancement of a smoothing estimates. Such upper bounds for the growth of Sobolev norms measure the transfer of energy from low to high modes as time grows on.

Energy‐critical scattering for focusing inhomogeneous coupled Schrödinger systems

This work investigates the time asymptotics of the inhomogeneous coupled Schrödinger equations , where , and . Here, one treats the energy‐critical regime and . This is the index of the invariant Sobolev norm under the dilatation . To the authors knowledge, the technique used in order to prove the scattering of an energy global solution to the above problem in the sub‐critical regime is no more applicable for . In order to overcome this difficulty, one uses the Kenig–Merle road map. In order to avoid a singularity of the source term, one considers the case , which restricts the space dimensions to . Moreover, in order to use the Sobolev injection , one restricts the space dimensions to . Compared with the previous work for the first author (Inhomogeneous coupled non‐linear Schrödinger systems. J. Math. Phys. 62, 101508 (2021)), the method consisting on dividing the integrals in the unit ball and its complementary seems not sufficient to conclude in the present study because of the energy‐critical exponent. Instead, one uses some Caffarelli–Kohn–Nirenberg weighted type inequalities.

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS. Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.

Well-posedness for the Schrödinger-KdV system on the half-line

In this paper we obtain improved local well–posedness results for the Schrödinger-KdV system on the half-line. We employ the Laplace–Fourier method in conjunction with the restricted norm method of Bourgain appropriately modified in order to accommodate the bounded operators of the half–line problem. Our result extends the previous local results in [5], [6] and [18] matching the results that Wu, [25], obtained for the real line system. We also demonstrate the uniqueness for the full range of locally well–posed solutions. In addition we obtain global well–posedness on the half–line for the energy solutions with zero boundary data, along with polynomial–in–time bounds for higher order Sobolev norms for the Schrödinger part.

On quantum Sobolev inequalities

We investigate the quantum analogue of the classical Sobolev inequalities in the phase space, with the quantum Sobolev norms defined in terms of Schatten norms of commutators. These inequalities provide an uncertainty principle for the Wigner–Yanase skew information, and also lead to new bounds on the Schatten norms of the Weyl quantization in terms of its symbol. As an intermediate tool, we obtain the analogue of Hardy–Littlewood–Sobolev's inequalities for a semiclassical analogue of the convolution, and introduce quantum Besov spaces. Explicit estimates are obtained on the optimal constants.

Fluctuations and moderate deviations for a catalytic Fleming–Viot branching system in nonequilibrium

We consider fluctuations and moderate deviations of the empirical fields for a catalytic Fleming–Viot branching system in nonequilibrium. We proved that for some independent initial distribution, the fluctuation process of the empirical fields is governed by an Ornstein–Uhlenbeck process whose drift term is a linear operator associated with a catalyst. Furthermore, we establish the large deviation principle corresponding to the fluctuation. We develop a technique to estimate the exponential moments for the Sobolev norms of the empirical fluctuation fields via the spectrum of the Laplace operator and the exponential inequality of martingales. The estimates of the exponential moments play a crucial role in this paper.

Data-driven learning of chaotic dynamical systems using Discrete-Temporal Sobolev Networks

We introduce the Discrete-Temporal Sobolev Network (DTSN), a neural network loss function that assists dynamical system forecasting by minimizing variational differences between the network output and the training data via a temporal Sobolev norm. This approach is entirely data-driven, architecture agnostic, and does not require derivative information from the estimated system. The DTSN is particularly well suited to chaotic dynamical systems as it minimizes noise in the network output which is crucial for such sensitive systems. For our test cases we consider discrete approximations of the Lorenz-63 system and the Chua circuit. For the network architectures we use the Long Short-Term Memory (LSTM) and the Transformer. The performance of the DTSN is compared with the standard MSE loss for both architectures, as well as with the Physics Informed Neural Network (PINN) loss for the LSTM. The DTSN loss is shown to substantially improve accuracy for both architectures, while requiring less information than the PINN and without noticeably increasing computational time, thereby demonstrating its potential to improve neural network forecasting of dynamical systems.

Effective chaos for the Kirchhoff equation on tori

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillate in a chaotic way on certain long timescales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, which we name effective chaos (it occurs over a long, but finite, timescale), is a consequence of the existence of symbolic dynamics for an effective system. Since the first-order dynamics has been proved to be essentially stable, we need to perform a second-order to find an effective model displaying chaotic dynamics. More precisely, after some nontrivial reductions, this model behaves as two weakly coupled pendulums.

A weakly turbulent solution to the cubic nonlinear harmonic oscillator on ℝ2 perturbed by a real smooth potential decaying to zero at infinity

We build a smooth real potential V(t, x) on ( t 0 , + ∞ ) × R 2 decaying to zero as t → ∞ and a smooth solution to the associated perturbed cubic noninear harmonic oscillator whose Sobolev norms blow up logarithmically as t → ∞ . Adapting the method of Faou and Raphael for the linear case, we modulate the Solitons associated to the nonlinear harmonic oscillator by time-dependent parameters solving a quasi-Hamiltonian dynamical system whose action grows up logarithmically, thus yielding logarithmic growth for the Sobolev norm of the solution.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied the in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behaviour of the NLS equation with random potential. More precisely, we assume that the amplitude of the potential is a random process whose law is 1 -periodic in time and nondegenerate. Combining the controllability with a stopping time argument and the Markov property, we show that the trajectories of the random equation are almost surely unbounded in regular Sobolev spaces.

Growth of Sobolev norms for 2d cubic NLS with partial harmonic potential

Growth of Sobolev norms for 2d cubic NLS with partial harmonic potential