Low Regularity Assumptions Research Articles

Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space RN with N≥3, on R+3 and on Ω⊆R3 exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation vu∈L∞((0,T);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)),∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for s>N2−1 if Ω=RN and s∈(12,1] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.

The cubic nonlinear fractional Schrödinger equation on the half-line

We study the cubic nonlinear fractional Schrödinger equation with Lévy indices 43<α<2 posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp in the sense of index of regularity required for the solutions with respect for known results on the full real line R. Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data and the corresponding linear solution. To get our results we use the Colliander and Kenig approach based on the Riemann–Liouville fractional operator combined with Fourier restriction method of Bourgain (1993) and some ideas of the recent work of Erdoğan et al. (2019). The method applies to both focusing and defocusing nonlinearities. As a consequence of our analysis we prove a smoothing effect for the cubic nonlinear fractional Schrödinger equation posed in full line R for the case of the low regularity assumption, which was point out at the recent work (Erdoğan et al., 2019).

Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations

&lt;p&gt;The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $ L^p $ initial vorticity, provided that $ p\geq 4 $. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order $ |\log\nu|^{-\alpha/2} $ with $ \alpha &amp;gt; 0 $.&lt;/p&gt;

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

Abstract We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal F_s$ . Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v&lt;s$ , and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal F_v$ by including a singularity in $(s-v)$ . We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

A fourth-order energy-preserving and symmetric average vector field integrator with low regularity assumption

It is known that the average vector field (AVF) integrator is energy-preserving but only with second-order accuracy. Motivated by the construction of the classical AVF integrator, we devoted to constructing and analysing of high-order energy-preserving integrators for the second-order Hamiltonian systems in this work. Firstly, a novel improved average vector field (IAVF) integrator is derived by modifying the average vector field of the AVF integrator. Then the energy preservation and symmetry of the derived IAVF integrator are rigorously studied. Another significant investigation of the paper is that the IAVF integrator is shown to converge with order four under low regularity assumption q(t)∈C3([t0,T]). Numerical experiments are conducted to demonstrate the remarkable superiority in accuracy and efficiency over the traditional AVF integrator. Numerical results confirm the conclusions of our theoretical analysis.

Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $$\mathcal {D}_t^\alpha (\partial _{t}u)$$ and $$\mathcal {D}_t^\alpha u $$ (with $$\alpha \in (0,1)$$ ), where $$\mathcal {D}_t^\alpha $$ denotes the Caputo fractional differential operator (in time) of order $$\alpha $$ . We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe’s method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multivariate Mittag-Leffler function.

On the antimaximum principle for the p-Laplacian and its sublinear perturbations

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ in a bounded domain $$\Omega \subset \mathbb {R}^N$$ , where $$q<p$$ . Under certain regularity and qualitative assumptions on the weights m, a and the source function f, we identify ranges of the parameters $$\lambda $$ and $$\eta $$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $$\eta =0$$ , and among them there are results providing new information even in the linear case $$p=2$$ . In particular, we show that for any $$p>1$$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the p-Laplacian provided $$m,f \in L^\gamma (\Omega )$$ with $$\gamma >N$$ . For completeness, we also investigate the existence of solutions.

General renewal equations motivated by biology and epidemiology

We present a unified framework ensuring well posedness and providing stability estimates to a class of Initial – Boundary Value Problems for renewal equations comprising a variety of biological or epidemiological models. This versatility is achieved considering fairly general – possibly non linear and/or non local – interaction terms, allowing both low regularity assumptions and independent variables with or without a boundary. In particular, these results also apply, for instance, to a model for the spreading of a Covid like pandemic or other epidemics. Further applications are shown to be covered by the present setting.

A C0 interior penalty method for mth-Laplace equation

In this paper, we propose a C0 interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝd, where m and d can be any positive integers. The standard H1-conforming piecewise r-th order polynomial space is used to approximate the exact solution u, where r can be any integer greater than or equal to m. Unlike the interior penalty method in Gudi and Neilan [IMA J. Numer. Anal. 31 (2011) 1734–1753], we avoid computing Dm of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete Hm-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete Hm-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate.

Global regularity for the 3D Hall-MHD equations with low regularity axisymmetric data

In this paper, we consider the global well-posedness of the incompressible Hall-MHD equations in \({\mathbb {R}}^3\). We prove that the solution of this system is globally regular if the initial data is axisymmetric and the swirl components of the velocity and magnetic vorticity are trivial. It should be pointed out that the initial data can be arbitrarily large and satisfy low regularity assumptions.

A new primal-dual weak Galerkin method for elliptic interface problems with low regularity assumptions

This article introduces a new primal-dual weak Galerkin (PDWG) finite element method for second order elliptic interface problems with ultra-low regularity assumptions on the exact solution and the interface and boundary data. It is proved that the PDWG method is stable and accurate with optimal order of error estimates in discrete and Sobolev norms. In particular, the error estimates are derived under the low regularity assumption of u∈Hδ(Ω) for δ>12 for the exact solution u. Extensive numerical experiments are conducted to provide numerical solutions that verify the efficiency and accuracy of the new PDWG method.

Nondiffusive variational problems with distributional and weak gradient constraints

Abstract In this article, we consider nondiffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a nonstandard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solution to this pre-dual problem under some assumptions. We conclude the article by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples illustrate the theoretical findings.

A new numerical method for div-curl systems with low regularity assumptions

This paper presents a new numerical method for div-curl systems with the normal boundary condition by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low Hα-regularity (α>0) assumption for the true solution, but also an effective approximation of the normal harmonic vector fields on domains with complex topology. Seven numerical experiments are conducted and the results are presented to demonstrate the performance of the PDWG algorithm, including one example on the computation of discrete normal harmonic vector fields.

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

An inverse source problem for a non-automonous time fractional diffusion equation of order \((0<\beta <1)\) is considered in a bounded Lipschitz domain in \(\mathbb {R}^d\). The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe’s method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from \(t = 0\). Rothe’s method on a uniform grid addresses the existence of a such a solution (non-smooth with \(t^\gamma \) term where \(1>\gamma > \beta \)) under low regularity assumptions, whilst Rothe’s method over graded grids has the advantage to cope better with the behaviour at \(t=0\) (also here \(t^\beta \) is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

SDEs with random and irregular coefficients

We consider Itô uniformly nondegenerate equations with random coefficients. When the coefficients satisfy some low regularity assumptions with respect to the spatial variables and Malliavin differentiability assumptions on the sample points, the unique solvability of singular SDEs is proved by solving backward stochastic Kolmogorov equations and utilizing a modified Zvonkin type transformation.

Rotational smoothing

Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes k-vector-space Riesz potentials in Rn with k<n, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on k-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in Rn, or a solution operator for the transport equation in Rn. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.

Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space

We consider a locally uniformly strictly elliptic second order partial differential operator in Rd, d≥2, with low regularity assumptions on its coefficients, as well as an associated Hunt process and semigroup. The Hunt process is known to solve a corresponding stochastic differential equation that is pathwise unique. In this situation, we study the relation of invariance, infinitesimal invariance, recurrence, transience, conservativeness and Lr-uniqueness, and present sufficient conditions for non-existence of finite infinitesimally invariant measures as well as finite invariant measures. Our main result is that recurrence implies uniqueness of infinitesimally invariant measures, as well as existence and uniqueness of invariant measures, both in subclasses of locally finite measures. We can hence make in particular use of various explicit analytic criteria for recurrence that have been previously developed in the context of (generalized) Dirichlet forms and present diverse examples and counterexamples for uniqueness of infinitesimally invariant, as well as invariant measures and an example where L1-uniqueness fails for one infinitesimally invariant measure but holds for another and pathwise uniqueness holds. Furthermore, we illustrate how our results can be applied to related work and vice versa.

Low regularity primal–dual weak Galerkin finite element methods for convection–diffusion equations

We consider finite element discretizations for convection–diffusion problems under low regularity assumptions. The derivation and analysis use the primal–dual weak Galerkin (PDWG) finite element framework. The Euler–Lagrange formulation resulting from the PDWG scheme yields a system of equations involving not only the equation for the primal variable but also its adjoint for the dual variable. We show that the proposed PDWG method is stable and convergent. We also derive a priori error estimates for the primal variable in the Hϵ-norm for ϵ∈[0,12). A series of numerical tests that validate the theory is presented as well.

Global solution for Massive Maxwell-Klein-Gordon equations with large Maxwell field

We derive the global dynamic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field) with a general class of data, in particular, for Maxwell field of arbitrary size, and by a gauge independent method. Due to the critical slow decay expected for the Maxwell field, the scalar field exhibits a loss of decay at the causal infinities within an outgoing null cone. To overcome the difficulty caused by such loss in the energy propagation, we uncover a hidden cancellation contributed by the Maxwell equation, which enables us to obtain the sharp control of the Maxwell field under a rather low regularity assumption on data. Our method can be applied to other physical field equations, such as the Einstein equations for which a similar cancellation structure can be observed.

A Hybridizable Discontinuous Galerkin Method for the Quad-Curl Problem

The quad-curl problem arises in magnetohydrodynamics, inverse electromagnetic scattering problems, and electromagnetic transmission eigenvalue problems. In this paper, we propose a hybridizable discontinuous Galerkin method to solve the quad-curl problem, and perform the analysis with a relatively low regularity assumption. The efficiency and accuracy of the proposed method are illustrated through numerical experiments.