arXiv subjects

Liviu I. Ignat

Publications and source records attributed to Liviu I. Ignat.

At least 19 recordsLinked to original sources

arXiv2026

Galerkin Approximation of the Fractional Hardy Constant

We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\geq 1$, with fractional exponent $s\in \left(0,\min\left\{1,\frac{N}{2}\right\}\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.

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arXiv2026

Optimal convergence rates for the finite element approximation of the Sobolev constant

We establish optimal convergence rates for the continuous piecewise affine finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p-Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.

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arXiv2026

Galerkin Approximation of the Fractional Sobolev Constant

We establish sharp estimates for the discrete optimal constant of the fractional Sobolev inequality in dimension $N\geq 1$, with fractional exponent $s\in (0,\min\{1,N/2\})$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in the unit ball, for which we employ a quasi-uniform and regular mesh.

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arXiv2025

Sharp numerical approximation of the Hardy constant

We study the $P_1$ finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in $\mathbb{R}^N$, for $N \geq 3$. Despite the fact that this constant is not attained in the associated Sobolev space $H^1$, our main result establishes an explicit, sharp, and dimension-independent rate of convergence proportional to $1/|\log h|^2$. The analysis carefully combines an improved Hardy inequality involving a reminder term with logarithmic weights, approximation estimates for Hardy-type singular radial functions constituting minimizing sequences, properties of piecewise linear and continuous finite elements, and weighted Sobolev space techniques. We also consider other closely related spectral problems involving the Laplacian with singular quadratic potentials obtaining sharp convergence rates.

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arXiv2024

The Hardy inequality and large time behaviour of the heat equation on $\mathbb{R}^{N-k}\times (0,\infty)^k$

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces $\mathbb{R}^{N-k}\times (0,\infty)^k$, $k\geq 0$. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in $L^2(\mathbb{R}^{N-k}\times (0,\infty)^k)$. Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case $k=0$, to any $k\geq 0$. We emphasize that the higher the value of $k$ the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

math.AP↗

arXiv2024

Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space

We study a non-local evolution equation on the hyperbolic space $\mathbb{H}^N$. We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions. We also consider a non-local, non-linear convection-diffusion equation on $\mathbb{H}^N$ governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brezis-Mironescu.

math.AP↗

arXiv2024

Finite element approximation of the Hardy constant

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.

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arXiv2023

Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution with this mass of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a ``projection'' of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.

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arXiv2023

Heat equations with fast convection: source-type solutions and large-time behaviour

We study the existence and uniqueness of source-type solutions to the Cauchy problem for the heat equation with fast convection under certain tail control assumptions. We allow the solutions to change sign, but we will in fact show that they have the same sign as the initial data, which is a multiple of the Dirac delta. As an application, we obtain the large-time behaviour of nonnegative bounded solutions with integrable initial data to heat equations with fast convection, covering the case of several dimensions that remained open since the end of last century.

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arXiv2022

A Convection-Diffusion model on a star shaped graph

In this paper we study a convection-diffusion equation on a star-shaped graph composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in C^1(\rr)$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $\partial_x(f(u(t, x))$ with $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$, we analyze the long time behavior of the solutions. For $q> 2$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case $q=2$, the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers' equation. Explicit representations of the limit profiles are obtained.

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arXiv2020

Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length

We study local (the heat equation) and nonlocal (convolution type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star shaped limit problem the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at $x = 0$ and initial datum $(2M/N)δ_{x=0}$ where M is the total mass of the initial condition for our original problem and $N$ is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit.

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arXiv2020

Hardy uniqueness principle for the linear Schrodinger equation on quantum regular trees

In this paper we consider the linear Schrodinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the real line with a piece-wise constant coefficient and uses this result in the context of regular trees. The second part treats the case of a LSE with a real potential in the framework of a star-shaped graph.

math.AP↗

arXiv2019

On the numerical approximations of the periodic Schrödinger equation

We consider semidiscrete finite differences schemes for the periodic Scrödinger equation in dimension one. We analyze whether the space-time integrability properties observed by Bourgain in the continuous case are satisfied at the numerical level uniformly with respect to the mesh size. For the simplest finite differences scheme we show that, as mesh size tends to zero, the blow-up in the $L^4$ time-space norm occurs, a phenomenon due to the presence of numerical spurious high frequencies. To recover the uniformity of this property we introduce two methods: a spectral filtering of initial data and a viscous scheme. For both of them we prove a $L^4$ time-space estimate, uniform with respect to the mesh size. {Warning 2019}: This paper was submitted to M2AN in 2007 and it was assigned the number 2007-29. It passed a first review round ( three reviews :-) ) without decision ("It is only after the consideration of a thoroughly revised version of your manuscript and a new iteration with all the referees that I will be in position to make my final decision.") After completing the PhD, I tried to publish the papers resulting from the thesis and I didn't spent too much time on other related problems (the periodic case for example). It was only recently that I discovered some interest in the subject from other authors and I decided to upload it on arxiv.org. Use with caution as there is no revision of the text in the last twelve years.

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arXiv2018

The Schrodinger Equation on a Star-Shaped Graph under General Coupling Conditions

We investigate dispersive and Strichartz estimates for the Schrödinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schrödinger equations.

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arXiv2018

Null Controllability of the Kuramoto-Sivashinsky Equation on star-shaped trees

In this paper we treat controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the involved equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one.

math.OC↗

arXiv2017

A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.

math.NA↗

arXiv2016

A splitting method for the augmented Burgers equation

In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions behave as the self-similar solutions of the viscous Burgers equation

math.NA↗

arXiv2015

A compactness tool for the analysis of nonlocal evolution equations

In this paper we give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times Ω)$ and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.

math.AP↗