Order Sobolev Research Articles

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<s≤32 is presented. Moreover, the local well-posedness of strong solutions in Sobolev space Hs(R) with s>32 is established by the pseudoparabolic regularization technique.

Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness

We present a variational approach aimed at enhancing the training of physics-informed neural networks (PINNs) and more general surrogate models for learning partial differential equations (PDE). In particular, we extend our formerly introduced notion of Sobolev cubatures to negative orders, enabling the approximation of negative order Sobolev norms. We mathematically prove the effect of negative order Sobolev cubatures in improving the condition number of discrete PDE learning problems, providing balancing scalars that mitigate numerical stiffness issues caused by loss imbalances. Additionally, we consider polynomial surrogate models (PSMs), which maintain the flexibility of PINN formulations while preserving the convexity structure of the PDE operators. The combination of negative order Sobolev cubatures and PSMs delivers well-conditioned discrete optimization problems, solvable via an exponentially fast convergent gradient descent for λ-convex losses. Our theoretical contributions are supported by numerical experiments, addressing linear and non-linear, forward and inverse PDE problems. These experiments show that the Sobolev cubature-based PSMs emerge as the superior state-of-the-art PINN technique.

Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds

In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the higher and fractional order Sobolev inequalities with the lower bounds. This extends to some extent the stability of the first order Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [18] to the higher and fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory. As another application of the stability of the HLS inequality, we also establish the stability of Beckner's [4] restrictive Sobolev inequalities of fractional order s with 0<s<n2 on the flat sub-manifold Rn−1 and the sphere Sn−1 with the explicit lower bound. When s=1, this implies the explicit lower bound for the stability of Escobar's first order Sobolev trace inequality [19] which has remained unknown in the literature.

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 the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.

Local Second Order Sobolev Regularity for p-Laplacian Equation in Semi-Simple Lie Group

In this paper, we establish a structural inequality of the ∞-subLaplacian ▵0,∞ in a class of the semi-simple Lie group endowed with the horizontal vector fields X1,…,X2n. When 1&lt;p≤4 with n=1 and 1&lt;p&lt;3+1n−1 with n≥2, we apply the structural inequality to obtain the local horizontal W2,2-regularity of weak solutions to p-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces R2n with n≥2, the range of this p obtained is already optimal.

Persistence of regularity of the solution to a hyperbolic boundary value problem in domain with corner

This article is about the question of the persistence of regularity for the solution to hyperbolic boundary value problem in the quarter-space. More precisely we will both consider the pure boundary value problem and the initial boundary value problem and we propose a functional space, based upon the high order Sobolev space in which a control of the data of the problem leads to a control of the solution (in the same space). The space proposed here contains the tangential Sobolev space. The analysis borrows some ideas of the study of characteristic boundary value problems in the half-space for which the good derivative to consider is known to be the tangential derivatives x1∂1 instead of the normal derivative ∂1. For quarter-space problems the good quantity to consider will be the radial derivative x1∂1+x2∂2 and then we recover the control of tangential derivatives x1∂1 and x2∂2 using explicit formulas in polar coordinates. The regularity of the solution is then established intrinsically by adapting the method introduced by the author to deal with half-space problems without using regularization methods.

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

We consider the non-linear eigenvalue equations characterizing Lp into Lq Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)2. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if 1p+1q=1. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

A systematic approach on the second order regularity of solutions to the general parabolic p-Laplace equation

We study a general form of a degenerate or singular parabolic equation ut-|Du|γ(Δu+(p-2)Δ∞Nu)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t-|Du|^{\\gamma }\\big (\\Delta u+(p-2)\\Delta _\\infty ^Nu\\big )=0 \\end{aligned}$$\\end{document}that generalizes both the standard parabolic p-Laplace equation and the normalized version that arises from stochastic game theory. We develop a systematic approach to study second order Sobolev regularity and show that D^2u exists as a function and belongs to L^2_text {loc} for a certain range of parameters. In this approach proving the estimate boils down to verifying that a certain coefficient matrix is positive definite. As a corollary we obtain, under suitable assumptions, that a viscosity solution has a Sobolev time derivative belonging to L^2_text {loc}.

Stability and decay rate of smooth solutions for the 3D Boussinesq equation with partial horizontal dissipation and thermal damping

In this paper, we investigate the stability and large time behavior of some three dimensional Boussinesq equations with the horizontal dissipation in the horizontal velocity components and thermal damping. Due to the lack of vertical dissipation in the horizontal velocity components and full dissipation in the vertical velocity component, these problems becomes very challenging. Combining the higher order Sobolev norm of the velocity and the temperature and decay estimates of the lower order Sobolev norm of the velocity and the temperature, the stability issue and precise decay rates have been built. The most difficulty is to estimate L1-norm of ‖∇hu‖L∞ in time variable t.

Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain

We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.

Realisations of elliptic operators on compact manifolds with boundary

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

Sparse optimization problems in fractional order Sobolev spaces

We consider optimization problems in the fractional order Sobolev spaces with sparsity promoting objective functionals containing L p -pseudonorms, . Existence of solutions is proven. By means of a smoothing scheme, we obtain first-order optimality conditions, which contain an equation with the fractional Laplace operator. An algorithm based on this smoothing scheme is developed. Weak limit points of iterates are shown to satisfy a stationarity system that is slightly weaker than that given by the necessary condition.

Positive radially symmetric ground states to a bi-harmonic problem

In this paper, we establish the existence of positive solutions to the semilinear problem for bi-harmonic operator $$ (-\Delta)^2u+u=|u|^{p-1}u, \ \ \text{ in }\ R^n. $$ One key tool is to use the Schwartz symmetrization method, in particular the Riesz inequality to this problem. The other tool is the generalized Ni-Strauss Lemma to radially symmetric functions in higher order Sobolev spaces.

Interpolation operator on negative Sobolev spaces

We introduce a Scott–Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method.

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.

Existence of a weak solution to a generalized Riemann-type hydrodynamical equation

In this paper, we study a generalized Riemann-type hydrodynamical equation and establish the existence of a weak solution to the equation in a lower order Sobolev space with by using a modified pseudo-parabolic regularization method. Although the absence of some important conservation laws leads to the failure of global existence of the approximate solutions, it has been addressed by making good use of the structure of the system.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

A variational approach to first order kinetic mean field games with local couplings

First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this article we construct global in time weak solutions to a first order mean field games system involving kinetic transport operators, where the costs are local (hence non-regularizing) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. This is achieved by characterizing solutions through two convex optimization problems in duality. As part of our approach, we develop tools for the analysis of mean field games on a non-compact domain by variational methods. We introduce a notion of ‘reachable set’, built from the initial measure, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L 1-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we prove higher order Sobolev estimates of the solutions.