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Analysis of PDEs

New submissions

[ total of 26 entries: 1-26 ]
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New submissions for Fri, 12 Apr 24

[1]  arXiv:2404.07232 [pdf, ps, other]
Title: Ideal Magnetohydrodynamics and Field Dislocation Mechanics
Authors: Amit Acharya
Subjects: Analysis of PDEs (math.AP); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph)

The fully nonlinear (geometric and material) system of Field Dislocation Mechanics is reviewed to establish an exact analogy with the equations of ideal magnetohydrodynamics (ideal MHD) under suitable physically simplifying circumstances. Weak solutions with various conservation properties have been established for ideal MHD recently by Faraco, Lindberg, and Szekelyhidi using the techniques of compensated compactness of Tartar and Murat and convex integration; by the established analogy, these results would seem to be transferable to the idealization of Field Dislocation Mechanics considered. A dual variational principle is designed and discussed for this system of PDE, with the technique transferable to the study of MHD as well.

[2]  arXiv:2404.07412 [pdf, ps, other]
Title: Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian
Comments: 18 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2403.08070
Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)

In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven. This conclusion is actually an interesting extension of F. Brock's classical result about the isoperimetric inequality for Steklov eigenvalues of the Laplacian given in the influential paper [Z. Angew. Math. Mech. 81 (2001) 69-71]. Besides, a related open problem has also been proposed in this paper.

[3]  arXiv:2404.07469 [pdf, ps, other]
Title: On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluid
Comments: 31 pages
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)

We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, $\Omega=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partial\Omega=\{|x|=1\}$. The existence and uniqueness of a spherically symmetric stationary solution, denoted as $(\tilde{\rho},\tilde{u})$, is first proved by I. Hashimoto and A. Matsumura in 2021. In this paper, we show that either $\tilde{\rho}$ is monotone increasing or $\tilde{\rho}$ attains a unique global minimum, and this is classified by the boundary condition of density. Moreover, we also derive a set of decay rates for $(\tilde{\rho},\tilde{u})$ which allows us to prove the long time stability of $(\tilde{\rho},\tilde{u})$ under small initial perturbations using the energy method. The main difficulty for this is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.

[4]  arXiv:2404.07531 [pdf, ps, other]
Title: Existence results for problems involving non local operator with an asymmetric weight and with a critical nonlinearity
Authors: Sana Benhafsia (LAMA), Rejeb Hadiji (LAMA)
Subjects: Analysis of PDEs (math.AP)

Recently, a great attention has been focused on the study of fractional and non-local operators of elliptic type, both for the pure mathematical research and in view of concrete real-world applications. We consider the following non local problem on $\mathbb{H}_0^s(\Omega) \subset L^{q_s}(\Omega)$, with $q_s :=\frac{2n}{n-2s}$, $s\in ]0, 1[$ and $n\geq 3$ \begin{equation}\int_{\mathbb{R}^n}p(x) \bigg(\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dy\bigg)dx-\lambda \int_\Omega |u(x)|^q dx, \qquad(1)\end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n, p :\mathbb{R}^n \to \mathbb{R}$ is a given positive weight presenting a global minimum $p_0 >0$ at $a \in \Omega$ and $\lambda$ is a real constant. In this work we show that for $q=2$ the infimum of (1) over the set $\{u\in \mathbb{H}_0^s(\Omega), ||u||_{L^{q_s}(\Omega)}=1\}$ does exist for some $k, s, \lambda$ and $n$ and for $q\geq 2$ we study non ground state solutions using the Mountain Pass Theorem.

[5]  arXiv:2404.07535 [pdf, ps, other]
Title: Free boundary regularity for the inhomogeneous one-phase Stefan problem
Subjects: Analysis of PDEs (math.AP)

In this paper, we prove that flat solutions to inhomogeneous one-phase Stefan problem are $C^{1,\alpha}$ in the $x_n$ direction.

[6]  arXiv:2404.07538 [pdf, ps, other]
Title: Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains
Comments: 23 pages
Subjects: Analysis of PDEs (math.AP)

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection-diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher P\'eclet numbers are also considered.

[7]  arXiv:2404.07592 [pdf, ps, other]
Title: Non-homogeneous fourth order elliptic inequalities with a convolution term motivated by the suspension bridge problem
Authors: Zhe Yu
Subjects: Analysis of PDEs (math.AP)

We are concerned with the study of the twin non-local inequalities featuring non-homogeneous differential operators $$\displaystyle -\Delta^2 u + \lambda\Delta u \geq (K_{\alpha, \beta} * u^p)u^q \quad\text{ in } \mathbb{R}^N (N\geq 1),$$ and $$\displaystyle \Delta^2 u - \lambda\Delta u \geq (K_{\alpha, \beta} * u^p)u^q \quad\text{ in } \mathbb{R}^N (N\geq 1),$$ with parameters $\lambda, p, q >0$, $0\leq \alpha \leq N$ and $\beta>\alpha-N$. In the above inequalities the potential $K_{\alpha,\beta}$ is given by $K_{\alpha, \beta}(x) = |x|^{-\alpha}\log^{\beta}(1 + |x|)$ while $K_{\alpha, \beta} * u^p$ denotes the standard convolution operator in $\mathbb{R}^N$. We discuss the existence and non-existence of non-negative solutions in terms of $N, p, q, \lambda, \alpha$ and $\beta$.

[8]  arXiv:2404.07631 [pdf, other]
Title: Lower semicontinuity and existence results for anisotropic TV functionals with signed measure data
Subjects: Analysis of PDEs (math.AP)

We study the minimization of anisotropic total variation functionals with additional measure terms among functions of bounded variation subject to a Dirichlet boundary condition. More specifically, we identify and characterize certain isoperimetric conditions, which prove to be sharp assumptions on the signed measure data in connection with semicontinuity, existence, and relaxation results. Furthermore, we present a variety of examples which elucidate our assumptions and results.

[9]  arXiv:2404.07737 [pdf, ps, other]
Title: Global regularity of 2D Rayleigh-Bénard equations with logarithmic supercritical dissipation
Comments: 18 pages
Subjects: Analysis of PDEs (math.AP)

In this paper, we study the global regularity problem for the 2D Rayleigh-B\'{e}nard equations with logarithmic supercritical dissipation. By exploiting a combined quantity of the system, the technique of Littlewood-Paley decomposition and Besov spaces, and some commutator estimates, we establish the global regularity of a strong solution to this equations in the Sobolev space $H^{s}(\mathbb{R}^{2})$ for $s \ge2$.

[10]  arXiv:2404.07749 [pdf, ps, other]
Title: Control of the Schrödinger equation in $\mathbb{R}^3$: The critical case
Comments: 20 pages. Comments are welcome
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)

This article deals with the $\dot{H}^{1}$--level exact controllability for the defocusing critical nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$. Firstly, we show the problem under consideration to be well-posed using Strichartz estimates. Moreover, through the Hilbert uniqueness method, we prove the linear Schr\"{o}dinger equation to be controllable. Finally, we use a perturbation argument and show local exact controllability for the critical nonlinear Schr\"{o}dinger equation.

[11]  arXiv:2404.07756 [pdf, ps, other]
Title: The limit as $s\nearrow 1$ of the fractional convex envelope
Comments: 14 pages
Subjects: Analysis of PDEs (math.AP)

We study the behavior of the fractional convexity when the fractional parameter goes to 1. For any notion of convexity, the convex envelope of a datum prescribed on the boundary of a domain is defined as the largest possible convex function inside the domain that is below the datum on the boundary. Here we prove that the fractional convex envelope inside a strictly convex domain of a continuous and bounded exterior datum converges when $s\nearrow 1$ to the classical convex envelope of the restriction to the boundary of the exterior datum.

[12]  arXiv:2404.07809 [pdf, ps, other]
Title: The Cattaneo-Christov approximation of Fourier heat-conductive compressible fluids
Subjects: Analysis of PDEs (math.AP)

We investigate the Navier-Stokes-Cattaneo-Christov (NSC) system in $\mathbb{R}^d$ ($d\geq3$), a model of heat-conductive compressible flows serving as a finite speed of propagation approximation of the Navier-Stokes-Fourier (NSF) system. Due to the presence of Oldroyd's upper-convected derivatives, the system (NSC) exhibits a \textit{lack of hyperbolicity} which makes it challenging to establish its well-posedness, especially in multi-dimensional contexts. In this paper, within a critical regularity functional framework, we prove the global-in-time well-posedness of (NSC) for initial data that are small perturbations of constant equilibria, uniformly with respect to the approximation parameter $\varepsilon>0$. Then, building upon this result, we obtain the sharp large-time asymptotic behaviour of (NSC) and, for all time $t>0$, we derive quantitative error estimates between the solutions of (NSC) and (NSF). To the best of our knowledge, our work provides the first strong convergence result for this relaxation procedure in the three-dimensional setting and for ill-prepared data.
The (NSC) system is partially dissipative and incorporates both partial diffusion and partial damping mechanisms. To address these aspects and ensure the large-time stability of the solutions, we construct localized-in-frequency perturbed energy functionals based on the hypocoercivity theory. More precisely, our analysis relies on partitioning the frequency space into \textit{three} distinct regimes: low, medium and high frequencies. Within each frequency regime, we introduce effective unknowns and Lyapunov functionals, revealing the spectrally expected dissipative structures.

[13]  arXiv:2404.07813 [pdf, ps, other]
Title: Illposedness of incompressible fluids in supercritical Sobolev spaces
Authors: Xiaoyutao Luo
Comments: 23 pages
Subjects: Analysis of PDEs (math.AP)

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0<s< \frac{5}{2} $, we construct a $C^\infty_c$ initial velocity field with arbitrarily small $H^{s} $ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equations for $0<s< \frac{1}{2} $, where $s = \frac{1}{2}$ is scaling critical for this equation.

[14]  arXiv:2404.07830 [pdf, ps, other]
Title: Global solution and singularity formation for the supersonic expanding wave of compressible Euler equations with radial symmetry
Subjects: Analysis of PDEs (math.AP)

In this paper, we define the rarefaction and compression characters for the supersonic expanding wave of the compressible Euler equations with radial symmetry. Under this new definition, we show that solutions with rarefaction initial data will not form shock in finite time, i.e. exist global-in-time as classical solutions. On the other hand, singularity forms in finite time when the initial data include strong compression somewhere. Several useful invariant domains will be also given.

[15]  arXiv:2404.07843 [pdf, ps, other]
Title: The search for NLS ground states on a hybrid domain: Motivations, methods, and results
Comments: Contribution for the Proceedings of the Summer School of the Puglia Trimester 2023 "Singularities, Asymptotics and Limiting Models". 17 pages, 1 figure. Keywords: hybrids, standing waves, nonlinear Schr\"odinger, ground states, delta interaction, radially symmetric solutions, rearrangements
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Functional Analysis (math.FA)

We discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schr\"odinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs.

Cross-lists for Fri, 12 Apr 24

[16]  arXiv:2404.07659 (cross-list from gr-qc) [pdf, ps, other]
Title: High-frequency solutions to the Einstein equations
Comments: 34 pages, review article
Subjects: General Relativity and Quantum Cosmology (gr-qc); Analysis of PDEs (math.AP)

We review recent mathematical results concerning the high-frequency solutions to the Einstein vacuum equations and the limits of these solutions. In particular, we focus on two conjectures of Burnett, which attempt to give an exact characterization of high-frequency limits of vacuum spacetimes as solutions to the Einstein-massless Vlasov system. Some open problems and future directions are discussed.

[17]  arXiv:2404.07660 (cross-list from math.FA) [pdf, ps, other]
Title: Approximation of Random Evolution Equations
Comments: 37 pages
Subjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Numerical Analysis (math.NA); Probability (math.PR)

In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretisation error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretisation in randomness. The main result are regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved. To this end, we derive error estimates for vector-valued PCE as well as a quantified version of the Trotter-Kato theorem for form-induced semigroups.

Replacements for Fri, 12 Apr 24

[18]  arXiv:2001.01662 (replaced) [pdf, other]
Title: Sharp bounds on the Nusselt number in Rayleigh-Bénard convection and a bilinear estimate via Carleson measures
Comments: The first version of the paper was withdrawn due to a problem with a scaling argument and in the proof of Proposition 3.1
Subjects: Analysis of PDEs (math.AP)
[19]  arXiv:2207.00863 (replaced) [pdf, ps, other]
Title: The Pogorelov estimates for degenerate curvature equations
Subjects: Analysis of PDEs (math.AP)
[20]  arXiv:2301.01992 (replaced) [pdf, other]
Title: Monotonicity of the period and positive periodic solutions of a quasilinear equation
Subjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
[21]  arXiv:2308.10760 (replaced) [pdf, ps, other]
Title: Liouville theorems and Harnack inequalities for Allen-Cahn type equation
Authors: Zhihao Lu
Subjects: Analysis of PDEs (math.AP)
[22]  arXiv:2311.09036 (replaced) [pdf, ps, other]
Title: Local near-field scattering data enables unique reconstruction of rough electric potentials
Comments: 33 pages. Preprint
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
[23]  arXiv:2401.17147 (replaced) [pdf, ps, other]
Title: Large data existence of global-in-time strong solutions to the incompressible Navier-Stokes equations in high space dimensions
Authors: Xiangsheng Xu
Subjects: Analysis of PDEs (math.AP)
[24]  arXiv:2403.16648 (replaced) [pdf, ps, other]
Title: On the Korteweg-de Vries limit for the Boussinesq equation
Comments: 17 pages, V1:Minor typos are corrected
Subjects: Analysis of PDEs (math.AP)
[25]  arXiv:2403.19020 (replaced) [pdf, other]
Title: The sticky particle dynamics of the 1D pressureless Euler-alignment system as a gradient flow
Comments: 30 pages, 1 figure
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
[26]  arXiv:2106.07282 (replaced) [pdf, ps, other]
Title: Heat kernel estimates for Markov processes of direction-dependent type
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)
[ total of 26 entries: 1-26 ]
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