Analysis of PDEs
New submissions
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New submissions for Thu, 30 May 24
- [1] arXiv:2405.18603 [pdf, ps, other]
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Title: A constant rank theorem for special Lagrangian equationsSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
Constant rank theorems are obtained for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation. The argument also leads to Liouville type results for the special Lagrangian equation with subcritical phase, matching the known rigidity results for semiconvex entire solutions to the quadratic Hessian equation.
- [2] arXiv:2405.18611 [pdf, ps, other]
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Title: A better bound on blow-up rate for the superconformal semilinear wave equationSubjects: Analysis of PDEs (math.AP)
We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a $|\log(T-t)|^q$ improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.
- [3] arXiv:2405.18612 [pdf, ps, other]
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Title: A semilinear problem associated to the space-time fractional heat equation in $\mathbb{R}^N$Subjects: Analysis of PDEs (math.AP)
We study the fully nonlocal semilinear equation $\partial_t^\alpha u+(-\Delta)^\beta u=|u|^{p-1}u$, $p\ge1$, where $\partial_t^\alpha$ stands for the Caputo derivative of order $\alpha\in (0,1)$ and $(-\Delta)^\beta$, $\beta\in(0,1]$, is the usual $\beta$ power of the Laplacian. We prescribe an initial datum in $L^q(\mathbb{R}^N)$.
We give conditions ensuring the existence and uniqueness of a solution living in $L^q(\mathbb{R}^N)$ up to a maximal existence time $T$ that may be finite or infinite. If~$T$ is finite, the $L^q$ norm of the solution becomes unbounded as time approaches $T$, and $u$ is said to blow up in $L^q$. Otherwise, the solution is global in time.
For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. It turns out that every nonnegative nontrivial solution in $L^q$ blows up in finite time if $1<p<p_f:=1+\frac{2\beta}N$ whereas if $p\ge p_f$ there are both solutions that blow up and global ones. The critical exponent $p_f$, which does not depend on $\alpha$, coincides with the Fujita exponent for the case $\alpha=1$, in which the time derivative is the standard (local) one. In contrast to the case $\alpha=1$, when $\alpha\in(0,1)$ the critical exponent $p=p_f$ falls within the situation in which global existence may occur. Our weakest condition for global existence and our condition for blow-up are both related to the size of the mean value of the initial datum in large balls. - [4] arXiv:2405.18673 [pdf, ps, other]
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Title: GAN: DynamicsComments: 28 pages, 3 figuresSubjects: Analysis of PDEs (math.AP)
We study quantitatively the overparametrization limit of the original Wasserstein-GAN algorithm. Effectively, we show that the algorithm is a stochastic discretization of a system of continuity equations for the parameter distributions of the generator and discriminator. We show that parameter clipping to satisfy the Lipschitz condition in the algorithm induces a discontinuous vector field in the mean field dynamics, which gives rise to blow-up in finite time of the mean field dynamics. We look into a specific toy example that shows that all solutions to the mean field equations converge in the long time limit to time periodic solutions, this helps explain the failure to converge.
- [5] arXiv:2405.18691 [pdf, other]
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Title: Partially invariant solution with an arbitrary surface of blow-up for the gas dynamics equations admitting pressure translationAuthors: Dilara SiraevaComments: 19 pages, 4 figuresSubjects: Analysis of PDEs (math.AP)
We applied a method of symmetry reduction to the gas dynamics equations with a special form of the equation of state. This equation of state is a pressure represented as the sum of a density and an entropy functions. The symmetry Lie algebra of the system is 12-dimensional. One, two and three-dimensional subalgebras were considered. In this article, four-dimensional subalgebras are considered for the first time. Specifically, invariants are calculated for 50 four-dimensional subalgebras. Using invariants of one of the subalgebras, a symmetry reduction of the original system is calculated. The reduced system is a partially invariant submodel because one gas-dynamic function cannot be expressed in terms of the invariants. The submodel leads to two families of exact solutions, one of which describes the isochoric motion of the media, and the other solution specifies an arbitrary blow-up surface. For the first family of solutions, the particle trajectories are parabolas or rays; for the second family of solutions, the particles move along cubic parabolas or straight lines. From each point of the blow-up surface, particles fly out at different speeds and end up on a straight line at any other fixed moment in time. A description of the motion of particles for each family of solutions is given.
- [6] arXiv:2405.18806 [pdf, other]
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Title: Propagation of Waves from Finite Sources Arranged in Line Segments within an Infinite Triangular LatticeComments: 22 pages, 15 figures, 2 tablesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
This paper examines the propagation of time harmonic waves through a two-dimensional triangular lattice with sources located on line segments. Specifically, we investigate the discrete Helmholtz equation with a wavenumber $k \in \left( 0,2\sqrt{2} \right)$, where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green's function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green's representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the left-handed 2D inductor-capacitor metamaterial.
- [7] arXiv:2405.18829 [pdf, other]
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Title: On the stationary solution of the Landau-Lifshitz-Gilbert equation on a nanowire with constant external magnetic fieldAuthors: Guillaume Ferriere (Paradyse)Subjects: Analysis of PDEs (math.AP)
We consider an infinite ferromagnetic nanowire, with an energy functional $E$ with easy-axis in the direction $e_1$ and a constant external magnetic field $E_{ext} = h_0 e_1$ along the same direction. The evolution of its magnetization is governed by the Landau-Lifshitz-Gilbert equation (LLG) associated to $E$. Under some assumptions on $h_0$, we prove the existence of stationary solutions with the same limits at infinity, their uniqueness up to the invariances of the equation and the instability of their orbits with respect to the flow. This property gives interesting new insights of the behavior of the solutions of (LLG), which are completed by some numerical simulations and discussed afterwards, in particular regarding the stability of 2-domain wall structures proven in [4] and more generally the interactions between domain walls.
- [8] arXiv:2405.18846 [pdf, ps, other]
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Title: One-dimensional boundary blow up problem with a nonlocal termSubjects: Analysis of PDEs (math.AP)
In this paper, we study a nonlocal boundary blow up problems on an interval and obtain the precise asymptotic formula for solutions when the bifurcation parameter in the problem is large.
- [9] arXiv:2405.18854 [pdf, ps, other]
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Title: Notes on asymptotic behavior of radial solutions for some weighted elliptic equations on the annulusAuthors: Futoshi TakahashiComments: 15 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we study the asymptotic behavior of radial solutions for several weighted elliptic equations with power type or exponential type nonlinearities on an annulus.
- [10] arXiv:2405.18943 [pdf, other]
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Title: Decoding a mean field game by the Cauchy data around its unknown stationary statesComments: Keywords: Mean field games, inverse problems, Cauchy data, unique continuation principle, unique identifiabilitySubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
In recent years, mean field games (MFGs) have garnered considerable attention and emerged as a dynamic and actively researched field across various domains, including economics, social sciences, finance, and transportation. The inverse design and decoding of MFGs offer valuable means to extract information from observed data and gain insights into the intricate underlying dynamics and strategies of these complex physical systems. This paper presents a novel approach to the study of inverse problems in MFGs by analyzing the Cauchy data around their unknown stationary states. This study distinguishes itself from existing inverse problem investigations in three key significant aspects: Firstly, we consider MFG problems in a highly general form. Secondly, we address the technical challenge of the probability measure constraint by utilizing Cauchy data in our inverse problem study. Thirdly, we enhance existing high order linearization methods by introducing a novel approach that involves conducting linearization around non-trivial stationary states of the MFG system, which are not a-priori known. These contributions provide new insights and offer promising avenues for studying inverse problems for MFGs. By unraveling the hidden structure of MFGs, researchers and practitioners can make informed decisions, optimize system performance, and address real-world challenges more effectively.
- [11] arXiv:2405.18954 [pdf, other]
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Title: Determining state space anomalies in mean field gamesComments: Keywords: Stationary mean field games, inverse boundary problems, anomalies in state space, singularities, uniquenessSubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
In this paper, we are concerned with the inverse problem of determining anomalies in the state space associated with the stationary mean field game (MFG) system. We establish novel unique identifiability results for the intrinsic structure of these anomalies in mean field games systems, including their topological structure and parameter configurations, in several general scenarios of practical interest, including traffic flow, market economics and epidemics. To the best of our knowledge, this is the first work that considers anomalies in the state space for the nonlinear coupled MFG system.
- [12] arXiv:2405.19025 [pdf, ps, other]
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Title: Fractional diffusion as the limit of a short range potential Rayleigh gasComments: 38 pages. arXiv admin note: text overlap with arXiv:2405.04449Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)
The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size $\varepsilon$ and the background is distributed in space $R^3$ according to a Poisson process with intensity $N$ and in velocity according some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as $\varepsilon$ tends to zero and $N$ tends to infinity with $N \varepsilon^2 =c$. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times $t \in [0,T]$, where $T$ and inverse mean free path $c$ can both be chosen as some negative rational power $\varepsilon^{-k}$.
- [13] arXiv:2405.19102 [pdf, other]
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Title: Annealed Calderón-Zygmund estimates for elliptic operators with random coefficients on $C^{1}$ domainsComments: 52pages; 2 figures; comments are welcome;Subjects: Analysis of PDEs (math.AP); Probability (math.PR)
Concerned with elliptic operators with stationary random coefficients governed by linear or nonlinear mixing conditions and bounded (or unbounded) $C^1$ domains, this paper mainly studies (weighted) annealed Calder\'on-Zygmund estimates, some of which are new even in a periodic setting. Stronger than some classical results derived by a perturbation argument in the deterministic case, our results own a scaling-invariant property, which additionally requires the non-perturbation method (based upon a quantitative homogenization theory and a set of functional analysis techniques) recently developed by M. Joisen and F. Otto \cite{Josien-Otto22}. To handle boundary estimates in certain UMD (unconditional martingale differences) spaces, we hand them over to Shen's real arguments \cite{Shen05, Shen23} instead of using Mikhlin's theorem. As a by-product, we also established ``resolvent estimates''. The potentially attractive part is to show how the two powerful kernel-free methods work together to make the results clean and robust.
- [14] arXiv:2405.19174 [pdf, ps, other]
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Title: Strong solution of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equationss with a modified dampingAuthors: Maroua LtifiSubjects: Analysis of PDEs (math.AP)
This study delves into a comprehensive examination of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equations in $H^{1}(\R^{3})$. The modification involves incorporating a power term in the nonlinear convection component, a particularly relevant adjustment in porous media scenarios, especially when the fluid adheres to the Darcy-Forchheimer law instead of the conventional Darcy law. Our main contributions include establishing global existence over time and demonstrating the uniqueness of solutions. It is important to note that these achievements are obtained with smallness conditions on the initial data, but under the condition that $\beta >3$ and $\alpha>0$. However, when $\beta=3$, the problem is limited to the case $0<\alpha<\frac{1}{2}$ as the above inequality is unsolvable for these values of $\alpha$ using our method. To support our statement, we will add a "slight disturbance" of the function f of the type $f(z)=log(e+z^{2})$ or $\log(\log(e^{e}+z^{2}))$ or even $\log(\log(\log((e^{e})^{e}+z^{2})))$.
- [15] arXiv:2405.19214 [pdf, ps, other]
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Title: Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spacesComments: 37 pages. arXiv admin note: text overlap with arXiv:2306.05948Subjects: Analysis of PDEs (math.AP)
In a previous work (arXiv:2306.05948), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on $\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$. In the present paper, we develop tools that significantly improve that result in two ways: Firstly, we achieve vorticities in $H^p(\mathbb{R}^2)$ in the optimal range $p\in (0,1)$ compared to $(2/3,1)$ in our previous work. Secondly, the solutions constructed here possess compact support and in particular preserve linear and angular momenta.
- [16] arXiv:2405.19233 [pdf, ps, other]
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Title: Pseudo-Gevrey Smoothing for the Passive Scalar Equations near CouetteComments: 130 pagesSubjects: Analysis of PDEs (math.AP)
In this article, we study the regularity theory for two linear equations that are important in fluid dynamics: the passive scalar equation for (time-varying) shear flows close to Couette in $\mathbb T \times [-1,1]$ with vanishing diffusivity $\nu \to 0$ and the Poisson equation with right-hand side behaving in similar function spaces to such a passive scalar. The primary motivation for this work is to develop some of the main technical tools required for our treatment of the (nonlinear) 2D Navier-Stokes equations, carried out in our companion work. Both equations are studied with homogeneous Dirichlet conditions (the analogue of a Navier slip-type boundary condition) and the initial condition is taken to be compactly supported away from the walls. We develop smoothing estimates with the following three features:
[1] Uniform-in-$\nu$ regularity is with respect to $\partial_x$ and a time-dependent adapted vector-field $\Gamma$ which approximately commutes with the passive scalar equation (as opposed to `flat' derivatives), and a scaled gradient $\sqrt{\nu} \nabla$;
[2] $(\partial_x, \Gamma)$-regularity estimates are performed in Gevrey spaces with regularity that depends on the spatial coordinate, $y$ (what we refer to as `pseudo-Gevrey');
[3] The regularity of these pseudo-Gevrey spaces degenerates to finite regularity near the center of the channel and hence standard Gevrey product rules and other amenable properties do not hold.
Nonlinear analysis in such a delicate functional setting is one of the key ingredients to our companion paper, \cite{BHIW24a}, which proves the full nonlinear asymptotic stability of the Couette flow with slip boundary conditions. The present article introduces new estimates for the associated linear problems in these degenerate pseudo-Gevrey spaces, which is of independent interest. - [17] arXiv:2405.19249 [pdf, ps, other]
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Title: Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary ConditionsComments: 157 pagesSubjects: Analysis of PDEs (math.AP)
We consider the 2D, incompressible Navier-Stokes equations near the Couette flow, $\omega^{(NS)} = 1 + \epsilon \omega$, set on the channel $\mathbb{T} \times [-1, 1]$, supplemented with Navier boundary conditions on the perturbation, $\omega|_{y = \pm 1} = 0$. We are simultaneously interested in two asymptotic regimes that are classical in hydrodynamic stability: the long time, $t \rightarrow \infty$, stability of background shear flows, and the inviscid limit, $\nu \rightarrow 0$ in the presence of boundaries. Given small ($\epsilon \ll 1$, but independent of $\nu$) Gevrey 2- datum, $\omega_0^{(\nu)}(x, y)$, that is supported away from the boundaries $y = \pm 1$, we prove the following results: \begin{align*} & \|\omega^{(\nu)}(t) - \frac{1}{2\pi}\int \omega^{(\nu)}(t) dx \|_{L^2} \lesssim \epsilon e^{-\delta \nu^{1/3} t}, & \text{(Enhanced Dissipation)} \\ & \langle t \rangle \|u_1^{(\nu)}(t) - \frac{1}{2\pi} \int u_1^{(\nu)}(t) dx\|_{L^2} + \langle t \rangle^2 \|u_2^{(\nu)}(t)\|_{L^2} \lesssim \epsilon e^{-\delta \nu^{1/3} t}, & \text{(Inviscid Damping)} \\ &\| \omega^{(\nu)} - \omega^{(0)} \|_{L^\infty} \lesssim \epsilon \nu t^{3+\eta}, \quad\quad t \lesssim \nu^{-1/(3+\eta)} & \text{(Long-time Inviscid Limit)} \end{align*} This is the first nonlinear asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries. The techniques we develop represent a major departure from prior works on nonlinear inviscid damping as physical space techniques necessarily play a central role. In this paper, we focus on the primary nonlinear result, while tools for handling the linearized parabolic and elliptic equations are developed in our separate, companion work.
- [18] arXiv:2405.19306 [pdf, ps, other]
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Title: Uniform-in-time estimates on the size of chaos for interacting Brownian particlesComments: 68 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)
We consider a system of classical Brownian particles interacting via a smooth long-range potential in the mean-field regime, and we analyze the propagation of chaos in form of sharp, uniform-in-time estimates on many-particle correlation functions. Our results cover both the kinetic Langevin setting and the corresponding overdamped Brownian dynamics. The approach is mainly based on so-called Lions expansions, which we combine with new diagrammatic tools to capture many-particle cancellations, as well as with fine ergodic estimates on the linearized mean-field equation, and with discrete stochastic calculus with respect to initial data. In the process, we derive some new ergodic estimates for the linearized Vlasov-Fokker-Planck kinetic equation that are of independent interest. Our analysis also leads to uniform-in-time concentration estimates and to a uniform-in-time quantitative central limit theorem for the empirical measure associated with the particle dynamics.
Cross-lists for Thu, 30 May 24
- [19] arXiv:2405.18619 (cross-list from math.NA) [pdf, ps, other]
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Title: Stability of the Rao-Nakra sandwich beam with a dissipation of fractional derivative type: theoretical and numerical studyComments: 23 pages, 14 figuresSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
This paper is devoted to the solution and stability of a one-dimensional model depicting Rao--Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fractional derivatives are substituted by diffusion equations relative to a new independent variable, $\xi$, resulting in an augmented model with a dissipative semigroup operator. Polynomial decay of energy is achieved, with a decay rate depending on the fractional derivative parameters. Both the polynomial decay and its dependency on the parameters of the generalized Caputo derivative are numerically validated. To this end, an energy-conserving finite difference numerical scheme is employed.
- [20] arXiv:2405.18947 (cross-list from math.FA) [pdf, ps, other]
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Title: On Structured Perturbations of Positive SemigroupsComments: 37 pagesSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP)
In this note we generalize perturbation results for positive $C_0$-semigroups on AM- and AL-spaces and give a Weiss--Staffans type perturbation result for generators of positive semigroups on Banach lattices. The abstract results are applied to domain perturbations of generators, a heat equation with boundary feedback and perturbations of the first derivative.
- [21] arXiv:2405.19064 (cross-list from math.DG) [pdf, ps, other]
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Title: Arnold-Thom conjecture for the arrival time of surfacesComments: 26 pagesSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
Following \L ojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove \L ojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in $\mathbb R^{n+1}$ with neck or non-degenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not $C^2.$ The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.
Replacements for Thu, 30 May 24
- [22] arXiv:2212.14757 (replaced) [pdf, ps, other]
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Title: Local Regularity of very weak $s$-harmonic functions via fractional difference quotientsComments: 24 pagesSubjects: Analysis of PDEs (math.AP)
- [23] arXiv:2306.02067 (replaced) [pdf, ps, other]
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Title: Vanishing dissipation limit for non-isentropic Navier-Stokes equations with shock dataComments: All comments are welcome!Subjects: Analysis of PDEs (math.AP)
- [24] arXiv:2404.07631 (replaced) [pdf, other]
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Title: Lower semicontinuity and existence results for anisotropic TV functionals with signed measure dataSubjects: Analysis of PDEs (math.AP)
- [25] arXiv:2310.00370 (replaced) [pdf, ps, other]
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Title: Characterizations of parabolic reverse Hölder classesComments: 35 pages. arXiv admin note: substantial text overlap with arXiv:2306.07600Subjects: Classical Analysis and ODEs (math.CA); Analysis of PDEs (math.AP)
- [26] arXiv:2311.01298 (replaced) [pdf, ps, other]
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Title: Almost holomorphic curves in real analytic hypersurfacesSubjects: Complex Variables (math.CV); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
- [27] arXiv:2311.04523 (replaced) [pdf, ps, other]
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Title: $L^p$-$L^q$ estimates for transition semigroups associated to dissipative stochastic systemsSubjects: Probability (math.PR); Analysis of PDEs (math.AP); Functional Analysis (math.FA)
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