Sobolev Regularity Research Articles

Low-regularity Seiberg–Witten moduli spaces on manifolds with boundary

For a compact spin^{c} manifold X with boundary b_1(partial X)=0, we consider moduli spaces of solutions to the Seiberg–Witten equations in a generalized double Coulomb slice in W^{1,2} Sobolev regularity. We prove they are Hilbert manifolds, prove denseness and “semi-infinite-dimensionality” properties of the restriction to partial X, and establish a gluing theorem. To achieve these, we prove a general regularity theorem and a strong unique continuation principle for Dirac operators, and smoothness of a restriction map to configurations of higher regularity on the interior, all of which are of independent interest.

Optimal Transport Approach to Sobolev Regularity of Solutions to the Weighted Least Gradient Problem

Optimal Transport Approach to Sobolev Regularity of Solutions to the Weighted Least Gradient Problem

On the failure of the chain rule for the divergence of Sobolev vector fields

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].

On echoes in magnetohydrodynamics with magnetic dissipation

We study the long time asymptotic behavior of the inviscid magnetohydrodynamic equations with magnetic dissipation near a combination of Couette flow and a constant magnetic field. Here we show that there exist nearby explicit global in time low frequency solutions, which we call waves. Moreover, the linearized problem around these waves exhibits resonances under high frequency perturbations, called echoes, which result in norm inflation in Gevrey regularity and infinite time blow-up in Sobolev regularity.

Möbius-invariant self-avoidance energies for non-smooth sets of arbitrary dimension and co-dimension

In the present paper we investigate generalizations of O'Hara's Möbius energy on curves [40], to Möbius-invariant energies on non-smooth subsets of Rn of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set Σ⊂Rn that locally finite energy implies that Σ is, in fact, an embedded Lipschitz submanifold of Rn – sometimes even smoother (depending on additional regularity assumptions on the admissible set). We also prove, on the other hand, that a local graph structure of low fractional Sobolev regularity on a set Σ is already sufficient to guarantee finite energy of Σ. This type of Sobolev regularity is exactly what one would expect in view of Blatt's characterization [5] of the correct energy space for the Möbius energy on closed curves. Our results hold in particular for Kusner and Sullivan's cosine energy EKS[37] since one of the energies considered here is equivalent to EKS.

On the trace of the wave group and regularity of potentials

For the wave equation $\partial_t^2-\Delta+V$ on $\mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $t\rightarrow 0$ for the relative trace of the wave group.

Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation

Abstract We carry on the investigation started in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3] about the regularity of weak solutions to the strongly degenerate parabolic equation u t - div ⁡ [ ( | D ⁢ u | - 1 ) + p - 1 ⁢ D ⁢ u | D ⁢ u | ] = f in ⁢ Ω T = Ω × ( 0 , T ) , u_{t}-\operatorname{div}\Bigl{[}(\lvert Du\rvert-1)_{+}^{p-1}\frac{Du}{\lvert Du% \rvert}\Bigr{]}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} for n ≥ 2 {n\geq 2} , p ≥ 2 {p\geq 2} and ( ⋅ ) + {(\,\cdot\,)_{+}} stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that f ∈ L loc p ′ ⁢ ( 0 , T ; B p ′ , ∞ , loc α ⁢ ( Ω ) ) , f\in L_{\mathrm{loc}}^{p^{\prime}}(0,T;B_{p^{\prime},\infty,\mathrm{loc}}^{% \alpha}(\Omega)), with α ∈ ( 0 , 1 ) {\alpha\in(0,1)} and p ′ = p / ( p - 1 ) {p^{\prime}=p/(p-1)} . This leads us to obtain higher fractional differentiability results for a function of the spatial gradient Du of the solutions. Moreover, we establish the higher summability of Du with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 2022, 2, Paper No. 125636], and on the other hand as the fractional version of some results established in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3].

Global Sobolev regular solution for Boussinesq system

Abstract This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) \Omega := {{\mathbb{R}}}^{2}\times \left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) \left({\bf{v}},\rho )\in {H}^{s}\left(\Omega )\times {{\mathbb{H}}}^{s}\left(\Omega ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) {H}^{s+2}\left(\Omega )\times {{\mathbb{H}}}^{s+2}\left(\Omega ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.

Sub-exponential stability for the beam equation

We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two categories of differentiability: (i) a subspace of Sobolev regularity, (ii) a subspace of infinitely many differentiable functions which is strictly contained in the Sobolev space but which strictly contains the Gevrey one. In both cases we prove exponential type times of stability. The result holds for almost all mass parameters and it is obtained by combining normal form techniques with a suitable Diophantine condition weaker than the one proposed by Bourgain. This is the first result of this kind in Sobolev regularity for a degenerate equation, where only one parameter is used to tune the linear frequencies of oscillations.

Global regularity of 2D Navier–Stokes free boundary with small viscosity contrast

This paper studies the dynamics of two incompressible immiscible fluids in two dimensions modeled by the inhomogeneous Navier–Stokes equations. We prove that if initially the viscosity contrast is small then there is global-in-time regularity. This result has been proved recently in Paicu and Zhang [Comm. Math. Phys. 376 (2020)] for H^{5/2} Sobolev regularity of the interface. Here we provide a new approach which allows us to obtain preservation of the natural C^{1+\gamma} Hölder regularity of the interface for all 0<\gamma<1 . Our proof is direct and allows for low Sobolev regularity of the initial velocity without any extra technicalities. It uses new quantitative harmonic analysis bounds for C^{\gamma} norms of even singular integral operators on characteristic functions of C^{1+\gamma} domains [Gancedo and García-Juárez, J. Funct. Anal. 283 (2022)].

Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds

Abstract In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner identity, and leads to “semilinear Calderón–Zygmund” type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed.

Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $$(-\Delta )^s$$ . We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $$C^s$$ up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases

We study the separation property for Cahn-Hilliard type equations with constant mobility and (physically relevant) singular potentials in two dimensions. That is, any solution with initial finite energy stays uniformly away from the pure phases $ \pm 1 $ from a certain time on. Beyond its physical interest, this property plays a crucial role to achieve high order Sobolev and analytic regularity of the solutions and to analyze their longtime behavior. In the local case, we streamline known arguments by exploiting the Sobolev inequality to obtain direct entropy estimates. In the nonlocal case, we provide a new proof based on De Giorgi estimates rather than the Alikakos-Moser type argument. Finally, in the spectral-fractional case, we prove nonlinear estimates and the separation property for any fractional index $ s\in (0,1) $ filling the gap between first-order (local) and zero-order (nonlocal) energy cases. In all of the aforementioned cases, our new proofs neither make use of the Trudinger-Moser inequality nor of any assumptions involving the third derivative of the entropy, as in the previous contributions. In particular, they apply for a more general class of singular potentials than the Flory-Huggins (Boltzmann-Gibbs) logarithmic density. Besides, the new methods present a series of technical advantages, which can be useful to the analysis of important physical systems that couple Cahn-Hilliard equations with other equations (e.g., reaction-diffusion equations and/or Navier-Stokes type systems) as well as their stochastic counterparts.

Sobolev regularity theory for the non-local elliptic and parabolic equations on $ C^{1,1} $ open sets

Sobolev regularity theory for the non-local elliptic and parabolic equations on $ C^{1,1} $ open sets

Global boundedness in a parabolic-parabolic-elliptic attraction-repulsion chemotaxis system with nonlinear productions

This paper investigates the global boundedness of the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: \begin{document}$ \begin{cases} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+u(a-bu^s), \ (x, t)\in \Omega\times (0, T), \\[6pt] v_t = \Delta v+\alpha u^k-\beta v, \ (x, t)\in \Omega\times (0, T), \\[6pt] 0 = \Delta w+\gamma u^l-\delta w, \ (x, t)\in\Omega\times (0, T), \end{cases} $\end{document} with $ \alpha, \beta, \gamma, \delta, \chi, \xi, a, b, s, k, l>0 $, subject to the homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\mathbb{R}^n $ ($ n\ge1 $) with smooth boundary. Hong et al. (J. Math. Anal. Appl. 484: 12703, 2020) have established the global boundedness criteria for the parabolic-elliptic-elliptic attraction-repulsion chemotaxis system. With the help of the maximum Sobolev regularity, this paper generalizes the results to the parabolic-parabolic-elliptic case. It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with $ \max\{l, s, \frac{2}{n}\}>k $, the solutions would be globally bounded. Under the balance situations, the boundedness of solutions will be determined by the sizes of the coefficients involved, but Hong et al only considered the balance of logistic source and repulsion mechanisms with $ \max\{l, s\} = k $, the balance produced by random diffusion was not taken into account, this paper consider the balance of random diffusion, logistic source and repulsion mechanisms with $ \max\{l, s, \frac{2}{n}\} = k $, the clear characterization of the global boundedness criteria is obtained for the parabolic-parabolic-elliptic attraction-repulsion chemotaxis system.

Sharp smoothing properties of averages over curves

Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$ , $d\ge 3$ . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $d\ge 3$ . As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma $ for $d\ge 4$ .

Deconvolution of spherical data corrupted with unknown noise

We consider the deconvolution problem for densities supported on a (d−1)-dimensional sphere with unknown center and unknown radius, in the situation where the distribution of the noise is unknown and without any other observations. We propose estimators of the radius, of the center, and of the density of the signal on the sphere that are proved consistent without further information. The estimator of the radius is proved to have almost parametric convergence rate for any dimension d. When d=2, the estimator of the density is proved to achieve the same rate of convergence over Sobolev regularity classes of densities as when the noise distribution is known.

Well-posedness for SQG sharp fronts with unbounded curvature

Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. Boundedness of curvature plays an important role in the theoretical [F. Gancedo and R. M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014) 635–639] and numerical [D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005) 5949–5952; R. K. Scott and D. G. Dritschel, Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system, Phys. Rev. Lett. 112 (2014) 144505] study of singularity formation. In this paper, we establish local well-posedness for SQG sharp fronts of low Sobolev regularity, [Formula: see text] for arbitrarily small [Formula: see text]. This is the first construction for SQG sharp front solutions with unbounded curvature.

Adaptive Estimation of a Function from its Exponential Radon Transform in Presence of Noise

In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a \(\log {n}\) factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.

Interior regularity results for inhomogeneous anisotropic quasilinear equations

We consider inhomogeneous p-Laplace type equations of the form $$-\textrm{div}\left( a(\nabla u)\right) =f$$ in a possibly anisotropic setting. Under general assumptions on the source term f, we obtain quantitative Sobolev regularity results for the stress field $$a(\nabla u)$$ and weighted $$L^2$$ estimates for the Hessian of the solution. As far as we know, our results are new or refine the ones available in literature also when restricted to the Euclidean setting.