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The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the stable case where $f(u)

We consider the top Lyapunov exponent associated to the advection-diffusion and linearised Navier-Stokes equations on the two-dimensional torus. The velocity field is given by the stochastic Navier-Stokes equations driven by a non-degenerate white-in-time noise with a power-law correlation structure. We show that the top Lyapunov exponent is bounded from below by a negative power of the diffusion parameter. This partially answers a conjecture of Doering and Miles and provides a first lower bound on the Batchelor scale in terms of the diffusivity. The proof relies on a robust analysis of the projective process associated to the linear equation, through its spectral median dynamics. We introduce a probabilistic argument to show that high-frequency states for the projective process are unstable under stochastic perturbations, leading to a Lyapunov drift condition and quantitative-in-diffusivity estimates., Comment: 51 Pages

The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of H\"ormander, following important prior contributions by G\"arding, Levitan, Avakumovi\'c, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a "pointwise Weyl law") for a single elliptic, self-adjoint operator. Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\overline{P}=(P_1,P_2,\dots, P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\leq i,j\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-\Delta)^\frac12$. We illustrate our result with several examples, including surfaces of revolution., Comment: 32 pages

We develop a new approach to the investigation of normalized solutions for nonlinear Schr\"odinger equations based on the analysis of the masses of ground states of the corresponding action functional. Our first result is a complete characterization of the masses of action ground states, obtained via a Darboux-type property for the derivative of the action ground state level. We then exploit this result to tackle normalized solutions with a twofold perspective. First, we prove existence of normalized nodal solutions for every mass in the $L^2$-subcritical regime, and for a whole interval of masses in the $L^2$-critical and supercritical cases. Then, we show when least energy normalized solutions/least energy normalized nodal solutions are action ground states/nodal action ground states., Comment: 21 pages

We derive the time-dependent von K\'arm\'an plate equations from three dimensional, purely atomistic particle models. In particular, we prove that a thin structure of interacting particles whose dynamics is governed by Newton's laws of motion is effectively described by the von K\'arm\'an equations in the limit of vanishing interatomic distance $\eps$ and vanishing plate thickness $h$. While the classical plate equations are obtained for $\eps \ll h \ll 1$, we find new plate equations for finitely many layers in the ultrathin case $\eps \sim h$.

The symmetrized Asymptotic Mean Value Laplacian $\tilde{\Delta}$, obtained as limit of approximating operators $\tilde{\Delta}_r$, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as $r \downarrow 0$, the operators $\tilde{\Delta}_r$ eventually admit isolated eigenvalues defined via min-max procedure on any compact locally Ahlfors regular metric measure space. Then we prove $L^2$ and spectral convergence of $\tilde{\Delta}_r$ to the Laplace--Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary., Comment: 35 pages, all comments welcome

The aim of this work is to analyse a shape optimization problem in a mechanical friction context. Precisely we perform a shape sensitivity analysis of a Tresca friction problem, that is, a boundary value problem involving the usual linear elasticity equations together with the (nonsmooth) Tresca friction law on a part of the boundary. We prove that the solution to the Tresca friction problem admits a directional shape derivative which moreover coincides with the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Then an explicit expression of the shape gradient of the Tresca energy functional is provided (which allows us to provide numerical simulations illustrating our theoretical results). Our methodology is not based on any regularization procedure, but rather on the twice epi-differentiability of the (nonsmooth) Tresca friction functional which is analyzed thanks to a change of variables which is well-suited in the two-dimensional case. The obstruction in the higher-dimensional case is discussed., Comment: 30 pages. arXiv admin note: text overlap with arXiv:2410.11750, arXiv:2410.12315

We study a Dirichlet problem driven by the (degenerate or singular) fractional $p$-Laplacian and involving a $(p-1)$-superlinear reaction at infinity, not necessarily satisfying the Ambrosetti-Rabinowitz condition. Using critical point theory, truncation, and Morse theory, we prove the existence of at least three nontrivial solutions to the problem., Comment: 18 pages

We discuss two optimization problems related to the fractional $p$-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional $p$-Laplacian with Dirichlet conditions, with a bounded weight function varying in a rearrangement class. Then, we investigate the maximization of the energy functional for general nonlinear equations driven by the same operator, as the reaction varies in a rearrangement class. In both cases, we provide a pointwise relation between the optimizing datum and the corresponding solution., Comment: 16 pages

In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schr\"{o}dinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^\infty{H}^\frac{1}{2}(I\times\T^3)$, then $u$ is global., Comment: arXiv admin note: text overlap with arXiv:2011.12925 by other authors

This paper is concerned with existence and uniqueness, and stability of the solution for the 3D Prandtl equation in a polynomial weighted Sobolev space. The main novelty of this paper is to directly prove the long time well-posdness to 3D Prandtl equation without any structural assumption by the energy method. Moreover, the solution's lifespan can be extended to any large $T$, provided that the initial data with a perturbation lies in the monotonic shear profile of small size $e^{-T}$. This result improves the work by Xu and Zhang (J. Differential Equations, 263(12)(2017), 8749-8803) on the 2D Prandtl equations, achieving an extension to the three-dimensional case., Comment: arXiv admin note: text overlap with arXiv:1511.04850 by other authors

We focus on the bilinear Strichartz estimates for free solutions to the Schr\"odinger equation on rescaled waveguides $\mathbb{R} \times \mathbb{T}_\lambda^n$, $\mathbb{T}_\lambda^n=(\lambda\mathbb{T})^n$ with $n\geq 1$ and their applications. First, we utilize a decoupling-type estimate originally from Fan-Staffilani-Wang-Wilson [Anal. PDE 11 (2018)] to establish a global-in-time bilinear Strichartz estimate with a `$N_2^\epsilon$' loss on $\mathbb{R} \times \mathbb{T}^n_\lambda$ when $n\geq1$, which generalize the local-in time estimate in Zhao-Zheng [SIAM J. Math. Anal. (2021)] and fills a gap left by the unresolved case in Deng et al. [J. Func. Anal. 287 (2024)]. Second, we prove the local-in-time angularly refined bilinear Strichartz estimates on the 2d rescaled waveguide $\mathbb{R} \times \mathbb{T}_\lambda$, which generalize the estimate obtained by Takaoka [J. Differ. Equa. 394 (2024)] with $\lambda=1$. As applications, we show the local well-posedness and small data scattering for nonlinear Schr\"odinger equations with algebraic nonlinearities in the critical space on $\mathbb R^m\times\mathbb{T}^n$ and the global well-posedness for cubic NLS on $\mathbb{R} \times \mathbb{T}$ in the lower regularity space $H^s$ with $s>\frac{1}{2}$., Comment: 40pages, 1figure

In this paper, we study a coupled Hartree-type system given by \[ \left\{ \begin{array}{ll} -\Delta u = K_{1}(x)(|x|^{-(N-\alpha)} * K_{1}(x)v^{2^{*}_{\alpha}})v^{2^{*}_{\alpha}-1} & \text{in } \mathbb{R}^N, \\[1mm] -\Delta v = K_{2}(x)(|x|^{-(N-\alpha)} * K_{2}(x)u^{2^{*}_{\alpha}})u^{2^{*}_{\alpha}-1} & \text{in } \mathbb{R}^N, \end{array} \right. \] which is critical with respect to the Hardy-Littlewood-Sobolev inequality. Here, $N \geq 5$, $\alpha < N - 5 + \frac{6}{N-2}$, $2^{*}_{\alpha} = \frac{N + \alpha}{N - 2}$, and $(x', x'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$. The functions $K_{1}(|x'|, x'')$ and $K_{2}(|x'|, x'')$ are bounded, nonnegative functions on $\mathbb{R}^{+} \times \mathbb{R}^{N-2}$, sharing a common, topologically nontrivial critical point. We address the challenge of establishing the nondegeneracy of positive solutions to the limiting system. By employing a finite-dimensional reduction technique and developing new local Poho\v{z}aev identities, we construct infinitely many synchronized-type solutions, with energies that can be made arbitrarily large.

In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the $p$-Laplacian, the $(p,q)$-Laplacian, the double phase operator, and the logarithmic double phase operator.

The research explores a novel mathematical model for closed loop insulin delivery systems, featuring a glucose sensitive membrane. It employs a sophisticated framework of nonlinear reaction diffusion equations and enzyme kinetics. Central to the study is the development of analytical solutions for the glucose, gluconic acid, and oxygen concentrations, which are meticulously validated against simulation outcomes. This validation underscores the model's accuracy in capturing the complex dynamics inherent in such systems. Additionally, the study leverages Akbar and Ganji's methodology to provide approximate solutions, enabling a comprehensive comparison with analytical results and offering deeper insights into the system's behavior under varying parameters. By integrating both analytical and approximate approaches, the research not only enhances our understanding of biochemical processes but also lays the groundwork for refining closed-loop insulin delivery technology. The findings promise to significantly improve the precision and efficacy of insulin administration, crucial for managing glucose levels in diabetic patients more effectively. Furthermore, the study's implications extend beyond insulin delivery, potentially informing the development of advanced biomedical systems where precise control and understanding of biochemical interactions are paramount. Ultimately, this work represents a significant contribution to both theoretical biochemistry and practical medical applications, setting a foundation for the next generation of closed-loop insulin delivery systems designed to better meet the complex metabolic needs of patients with diabetes., Comment: 27 pages, 23 figures

For a heterogeneous host population, the basic reproduction number of an infectious disease, $\cR_0$, is defined as the spectral radius of the next generation operator (NGO). The threshold properties of the basic reproduction number are typically established by imposing conditions that make $\cR_0$ an eigenvalue of the NGO associated with a positive eigenvector and a positive eigenfunctional (eigenvector of the dual of the NGO). More general results can be obtained by imposing conditions that associate $\cR_0$ just with a positive eigenfunctional. The next generation operator is conveniently expressed by a measure kernel or a Feller kernel which enables the use of analytic rather than functional analytic methods., Comment: 47 pages

We study the inverse problem of determining a time-dependent globally hyperbolic Lorentzian metric from the scattering operator for semilinear wave equations., Comment: 62 pages

In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fr\'echet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets., Comment: 19 pages

The Ohta-Kawasaki equation models the mesoscopic phase separation of immiscible polymer chains that form diblock copolymers, with applications in directed self-assembly for lithography. We perform a mathematical analysis of this model under degenerate mobility and an external force, proving the existence of weak solutions via an approximation scheme for the mobility function. Additionally, we propose a fully discrete scheme for the system and demonstrate the existence and uniqueness of its discrete solution, showing that it inherits essential structural-preserving properties. Finally, we conduct numerical experiments to compare the Ohta-Kawasaki system with the classical Cahn-Hilliard model, highlighting the impact of the repulsion parameter on the phase separation dynamics.

This work investigates a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.

We analyze a quasi-incompressible Cahn--Hilliard--Navier--Stokes system (qCHNS) for two-phase flows with unmatched densities. The order parameter is the volume fraction difference of the two fluids, while mass-averaged velocity is adopted. This leads to a quasi-incompressible model where the pressure also enters the equation of the chemical potential. We establish local existence and uniqueness of strong solutions by the Banach fixed point theorem and the maximal regularity theory., Comment: 29 pages

Conformally invariant scalar equation permits the Robin boundary condition at infinity of asymptotically-AdS spacetimes. We show how the dynamics of conformal cubic scalar field on the Reissner-Nordstr\"om-anti-de Sitter background depend on the black hole size, charge and the choice of the boundary condition. We study the whole range of admissible charges, including the extremal case. In particular, we observe the transition in stability of the field for large black holes at the specific critical value of the charge. Similarities between Reissner-Nordstr\"om and Kerr black hole let us suspect that similar effect may also occur in rotating black holes., Comment: 13 pages, 8 figures

In this article we study a reaction diffusion system with $m$ unknown concentration. The non-linearity in our study comes from an underlying reversible chemical reaction and triangular in nature. Our objective is to understand the large time behaviour of solution where there are degeneracies. In particular we treat those cases when one of the diffusion coefficient is zero and others are strictly positive. We prove convergence to equilibrium type of results under some condition on stoichiometric coefficients in dimension $1$,$2$ and $3$ in correspondence with the existence of classical solution. For dimension greater than 3 we prove similar result under certain closeness condition on the non-zero diffusion coefficients and with the same condition imposed on stoichiometric coefficients. All the constant occurs in the decay estimates are explicit., Comment: 45 pages, comments welcome

We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $\Omega\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(\Omega) \bigcap L^s(\Omega)$ $$\|\Delta u\|_{2} \leq \begin{cases} c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_q^2\|f\|_s, & \text{if } 1 < s < d/2, \frac{1}{2}=\frac{2}{q}+ \frac{1}{s} - \frac{2}{d}\\ c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_4^2\|f\|_s, & \text{if } s > d/2 \end{cases}.$$, Comment: 9 pages

We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure and the conformal type of the manifold. Moreover, the metric and the coefficient of the non-linearity are determined up to a multiplicative transformation. The manifold on which the inverse problem is considered is allowed to be an open, globally hyperbolic manifold which may have bifurcate event horizons or several infinities (i.e., ends) of which at least one has to be of the asymptotically Minkowskian type. The results are applied also for FLRW space-times that have no particle horizons. To formulate the inverse problems we define a new type of data, non-linear scattering functionals, which are defined also in the cases when the classically defined scattering operator is not well defined. This makes it possible to solve the inverse problems also in the case when some of the incoming waves lead to a blow-up of the scattered solution., Comment: 63 pages, 9 figures

We consider the initial-boundary value problem in the quarter space for the system of equations of ideal Magneto-Hydrodynamics for compressible fluids with perfectly conducting wall boundary conditions. On the two parts of the boundary the solution satisfies different boundary conditions, which make the problem an initial-boundary value problem with non-uniformly characteristic boundary. We identify a subspace ${{\mathcal H}}^3(\Omega)$ of the Sobolev space $H^3(\Omega)$, obtained by addition of suitable boundary conditions on one portion of the boundary, such that for initial data in ${{\mathcal H}}^3(\Omega)$ there exists a solution in the same space ${{\mathcal H}}^3(\Omega)$, for all times in a small time interval. This yields the well-posedness of the problem combined with a persistence property of full $H^3$-regularity, although in general we expect a loss of normal regularity near the boundary. Thanks to the special geometry of the quarter space the proof easily follows by the "reflection technique".

In this paper, we deal with the following mixed local/nonlocal Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{ll} - \Delta u + (-\Delta)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,} u>0 \quad \hbox{in $\mathbb{R}^n$,} \lim\limits_{|x|\to+\infty}u(x)=0, \end{array} \right. \end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$. The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions. The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- \Delta + (-\Delta)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques., Comment: 53 pages. To appear in Discrete and Continuous Dynamical Systems

Let $p\ge 1$ and let $\mathbf v \colon \mathbb R^d \to \mathbb R^d$ be a compactly supported vector field with $\mathbf v \in L^p(\mathbb R^d)$ and $\operatorname{div} \mathbf v = 0$ (in the sense of distributions). It was conjectured by Nelson that it $p=2$ then the operator $\mathsf{A}(\rho) := \mathbf v \cdot \nabla \rho$ with the domain $D(\mathsf A)=C_0^\infty(\mathbb R^d)$ is essentially skew-adjoint on $L^2(\mathbb R^d)$. A counterexample to this conjecture for $d\ge 3$ was constructed by Aizenmann. From recent results of Alberti, Bianchini, Crippa and Panov it follows that this conjecture is false even for $d=2$. Nevertheless, we prove that for $d=2$ the condition $p\ge 2$ is necessary and sufficient for the following chain rule property of $\mathbf v$: for any $\rho \in L^\infty(\mathbb R^2)$ and any $\beta\in C^1(\mathbb R)$ the equality $\operatorname{div}(\rho \mathbf v) = 0$ implies that $\operatorname{div}(\beta(\rho) \mathbf v) = 0$. Furthermore, for $d=2$ we prove that $\mathbf v$ has the renormalization property if and only if the stream function (Hamiltonian) of $\mathbf v$ has the weak Sard property, and that both of the properties are equivalent to uniqueness of bounded weak solutions to the Cauchy problem for the corresponding continuity equation. These results generalize the criteria established for $d=2$ and $p=\infty$ by Alberti, Bianchini and Crippa., Comment: 30 pages, 3 figures

A particular instance of the inverse magnetisation problem is considered. It is assumed that the support of a magnetic sample (a source term in the Poisson equation in $\mathbb{R}^3$) is contained in a bounded planar set parallel to the measurement plane. Moreover, only one component of the magnetic field is assumed to be known (measured) over the same planar region in the measurement plane. We propose a method to extrapolate the measurement data to the whole plane relying on the knowledge of the forward operator and the geometry of the problem. The method is based on the spectral decomposition of an auxiliary matrix-function operator. The results are illustrated numerically.

This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities. We then prove the interior $C^{1,\alpha}$-regularity and the $C^{1,\alpha}$-regularity up to the boundary of weak solutions, which extends previous results by the authors [X. Su, E. Valdinoci, Y. Wei and J. Zhang, Math. Z. (2022)], where the nonlinearities considered were of subcritical type. In addition, we establish the interior $C^{2,\alpha}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,\alpha}$-regularity up to the boundary for all $s\in(0,\frac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue., Comment: Journal of Differential Equations

In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for developing numerical reconstruction algorithm for this inverse problem. For this purpose, we develop a new method, based on different properties of solutions of elliptic equations, for treating the determination of the semilinear term as a source term from a point measurement of the solutions. This approach not only allows us to make important relaxations on the data used so far for solving this class of inverse problems, including general Dirichlet excitation lying in a space of dimension one and measurements located at one point on the boundary of the domain, but it also allows us to derive a novel algorithm for the reconstruction of the semilinear term. The effectiveness of our algorithm is corroborated by extensive numerical experiments. Notably, as demonstrated by the theoretical analysis, we are able to effectively reconstruct the unknown nonlinear source term by utilizing solely the information provided by the measurement data at a single point.

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show that if $F(x, u)$ is monomial and $G(x, u)$ is analytic in $u$, then $F, G$ and $c$ as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted of applications with nonlinear waves.

This paper introduces a novel mathematical framework for examining the regularity and energy dissipation properties of solutions to the stochastic Navier-Stokes equations. By integrating Sobolev-Besov hybrid spaces, fractional differential operators, and quantum-inspired modeling techniques, we provide a comprehensive analysis that captures the multiscale and chaotic dynamics inherent in turbulent flows. Central to this framework is a Schr\"odinger-type operator adapted for fluid dynamics, which encapsulates quantum-scale turbulence effects, thereby elucidating the mechanisms of energy redistribution across scales. Additionally, we develop anisotropic stochastic models with direction-dependent viscosity, characterized by a pseudo-differential operator and a covariance matrix governing directional diffusion. These models more accurately reflect real-world turbulence, where viscosity varies with flow orientation, enhancing both theoretical insights and practical simulation capabilities. Our main contributions include new regularity theorems and rigorous a priori estimates for solutions in Sobolev-Besov spaces, alongside proofs of energy dissipation properties in anisotropic contexts. These findings advance the understanding of fluid turbulence by offering a refined approach to studying scale interactions, stochastic effects, and anisotropy in turbulent flows., Comment: 12 pages

The goal of this note is to consider Liouville type theorem for p-Laplacian type operators. In particular guided by the Laplacian case one establishes analogous results for the p-Laplacian and operators of this type., Comment: Key words: p-Laplace operator, Liouville theorem, Schr\"odinger equation, nonlinear operators, anisotropic Laplace operator, double phase problem

We consider Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\text{MMD}_K^2(\cdot, \nu)$ for positive and negative distance kernels $K(x,y) := \pm |x-y|$ and given target measures $\nu$ on $\mathbb{R}$. Since in one dimension the Wasserstein space can be isometrically embedded into the cone $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions, Wasserstein gradient flows can be characterized by the solution of an associated Cauchy problem on $L_2(0,1)$. While for the negative kernel, the MMD functional is geodesically convex, this is not the case for the positive kernel, which needs to be handled to ensure the existence of the flow. We propose to add a regularizing Sobolev term $|\cdot|^2_{H^1(0,1)}$ corresponding to the Laplacian with Neumann boundary conditions to the Cauchy problem of quantile functions. Indeed, this ensures the existence of a generalized minimizing movement for the positive kernel. Furthermore, for the negative kernel, we demonstrate by numerical examples how the Laplacian rectifies a "dissipation-of-mass" defect of the MMD gradient flow., Comment: 27 pages, 13 figures

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent H\"{o}lder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of H\"{o}lder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.

We investigate the spectral stability of non-degenerate vector soliton solutions and the nonlinear stability of breather solutions for the coupled nonlinear Schrodinger (CNLS) equations. The non-degenerate vector solitons are spectrally stable despite the linearized operator admits either embedded or isolated eigenvalues of negative Krein signature. The nonlinear stability of breathers is obtained by the Lyapunov method with the help of the squared eigenfunctions due to integrability of the CNLS equations., Comment: 59 pages

We provide a new existence result for abstract nonlinear operator systems in normed spaces, by means of topological methods. The solution is located within the product of annular regions and conical shells. The theoretical result possesses a wide range of applicability, which, for concreteness, we illustrate in the context of systems of nonlinear Poisson equations subject to homogeneous Dirichlet boundary conditions. For the latter problem we obtain existence and localization of solutions having all components nontrivial. This is also illustrated with an explicit example in which we also furnish a numerically approximated solution, consistent with the theoretical results., Comment: 18 pages, 2 figures

In this paper, we study the inverse problems of simultaneously determining a potential term and a time-dependent nonlinearity for the nonlinear Moore-Gibson-Thompson equation with a fractional Laplacian. This nonlocal equation arises in the field of peridynamics describing the ultrasound waves of high amplitude in viscous thermally fluids. We first show the well-posedness for the considered nonlinear equations in general dimensions with small exterior data. Then, by applying the well-known linearization approach and the unique continuation property for the fractional Laplacian, the potential and nonlinear terms are uniquely determined by the Dirichlet-to-Neumann (DtN) map taking on arbitrary subsets of the exterior in the space-time domain. The uniqueness results for general nonlinearities to some extent extend the existing works for nonlocal wave equations. Particularly, we also show a uniqueness result of determining a time-dependent nonlinear coefficient for a 1-dimensional fractional Jordan-Moore-Gibson-Thompson equation of Westervelt type.

This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive - in an appropriate scaling regime - a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers.

This paper investigates the long time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of $N$ interacting point particles, where the time evolution of the joint distribution $f_N(t)$ is governed by the Liouville equation. Our primary objective is to analyze the system's behavior over extended time intervals, focusing on stability, potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime where $N \gg 1$, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases it is possible to derive uniform bounds in $ L^2 $, both over time and with respect to the number of particles, on the marginals $ \left(f_{k,N}\right)_{1\leq k \leq N}$, irrespective of the initial state's chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels surpassing the traditional $L^d$ regularity barriers, $K \in W^{\frac{-2}{d+2},d+2}(\mathbb{T}^d)$, where $\mathbb{T}$ denotes the $1$-torus and $d\geq2$ is the dimension. Finally, we address the highly singular case of $K \in H^{-1}(\mathbb{T}^d)$ within high-temperature regimes, offering new insights into the behavior of such systems.

We continue our investigation of Hardy-type inequalities involving combinations of cylindrical and spherical weights. Compared to [Cora-Musina-Nazarov, Ann. Sc. Norm. Sup., 2024], where the quasi-spherical case was considered, we handle the full range of allowed parameters. This has led to the observation of new phenomena related to lack of compactness., Comment: 30 pages

In this paper we consider the existence of standing waves for a coupled system of $k$ equations with Lotka-Volterra type interaction. We prove the existence of a standing wave solution with all nontrivial components satisfying a prescribed asymptotic profile. In particular, the $k-1$-last components of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. We analyze first in detail the result with three equations since this is the first case in which the coupling has a role contrary to what happens when only two densities appear. We also discuss the existence of solutions of this form for systems with other kind of couplings making a comparison with Lotka-Volterra type systems.

Evolutionary competition often occurs simultaneously at multiple levels of organization, in which traits or behaviors that are costly for an individual can provide collective benefits to groups to which the individual belongs. Building off of recent work that has used ideas from game theory to study evolutionary competition within and among groups, we study a PDE model for multilevel selection that considers group-level evolutionary dynamics through a pairwise conflict depending on the strategic composition of the competing groups. This model allows for incorporation of group-level frequency dependence, facilitating the exploration for how the form of probabilities for victory in a group-level conflict can impact the long-time support for cooperation via multilevel selection. We characterize well-posedness properties for measure-valued solutions of our PDE model and apply these properties to show that the population will converge to a delta-function at the all-defector equilibrium when between-group selection is sufficiently weak. We further provide necessary conditions for the existence of bounded steady state densities for the multilevel dynamics of Prisoners' Dilemma and Hawk-Dove scenarios, using a mix of analytical and numerical techniques to characterize the relative strength of between-group selection required to ensure the long-time survival of cooperation via multilevel selection. We also see that the average payoff at steady state appears to be limited by the average payoff of the all-cooperator group, even for games in which groups achieve maximal average payoff at intermediate levels of cooperation, generalizing behavior that has previously been observed in PDE models of multilevel selection with frequency-indepdent group-level competition., Comment: 60 pages. 12 figures

We establish the asymptotic stability of the catenoid, as a nonflat stationary solution to the hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space $\mathbb{R}^{1 + (n + 1)}$ for $n = 4$. Our main result is under a ``codimension-$1$'' assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \geq 5$ case addressed by L\"{u}hrmann-Oh-Shahshahani arxiv:2212.05620, proving catenoid stability in $4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties in the $n = 3$ case, the strong Huygens principle, as well as a miraculous cancellation in the source term, plays an important role in arxiv:2409.05968 to obtain strong late time tails. In $n = 4$ dimensions, without these special structural advantages, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy of estimates with higher $r^p$-weights so that an improved pointwise decay can be established. We expect this to be applicable for proving improved late time tails of other quasilinear wave equations in even dimensions or wave equations with inverse square potential., Comment: 105 pages, 1 figure

In this paper, for a family of second-order parabolic system or equation with rapidly oscillating and time-dependent periodic coefficients over rough boundaries, we obtain the large-scale boundary estimates, by a quantitative approach. The quantitative approach relies on approximating twice: we first approximate the original parabolic problem over rough boundary by the same equation over a non-oscillating boundary and then approximate the oscillating equation over a non-oscillating boundary by its homogenized equation over the same non-oscillating boundary.

The role of memory and cognition in the movement of individuals (e.g. animals) within a population, is thought to play an important role in population dispersal. In response, there has been increasing interest in incorporating spatial memory effects into classical partial differential equation (PDE) models of animal dispersal. However, the specific detail of the transport terms, such as diffusion and advection terms, that ought to be incorporated into PDE models to accurately reflect the memory effect remains unclear. To bridge this gap, we propose a straightforward lattice-based model where the movement of individuals depends on both crowding effects and the historic distribution within the simulation. The advantage of working with the individual-based model is that it is straightforward to propose and implement memory effects within the simulation in a way that is more biologically intuitive than simply proposing heuristic extensions of classical PDE models. Through deriving the continuum limit description of our stochastic model, we obtain a novel nonlinear diffusion equation which encompasses memory-based diffusion terms. For the first time we reveal the relationship between memory-based diffusion and the individual-based movement mechanisms that depend upon memory effects. Through repeated stochastic simulation and numerical explorations of the mean-field PDE model, we show that the new PDE model accurately describes the expected behaviour of the stochastic model, and we also explore how memory effects impact population dispersal., Comment: 9 figures

We study the stability of shock profiles in one spatial dimension for the isothermal Navier-Stokes-Poisson (NSP) system, which describes the dynamics of ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling waves, called shock profiles, for a given far-field condition satisfying the Lax entropy condition. In this paper, we prove that if the initial data is sufficiently close to a shock profile in $H^2$-norm, then the global solution of the Cauchy problem tends to the smooth manifold formed by the parametrized shock profiles as time goes to infinity. This is achieved using the method of $a$-contraction with shifts, which does not require the zero mass condition., Comment: 44 pages

We present a variation of the Front Tracking (FT) method for modeling solutions to hyperbolic systems in one space dimension, a Modified FT scheme. Instead of using non-entropic shocks, we approximate simple waves by jumps which exactly match the states, while approximating the wave speed. Our construction makes use of compression curve as well. We work with weak* solutions introduced in \cite{}. Consequently, we are able to analyze residuals rather than errors, and obtain cleaner convergence results. Through this paper, we intend to set up a scheme that is capable of handling strong shocks consistently. We develop this scheme primarily to prove (in a future work) existence of large amplitude solutions of the p-system. Therefore, we treat the p-system more carefully and construct a completely self-sufficient scheme, which has a limit as long as the approximate weak* solutions have uniformly bounded variation., Comment: 43 pages

We study the long-time behaviour of solutions to some classes of fourth-order nonlinear PDEs with non-monotone nonlinearities, which include the Landau--Lifshitz--Baryakhtar (LLBar) equation (with all relevant fields and spin torques) and the convective Cahn--Hilliard/Allen--Cahn (CH-AC) equation with a proliferation term, in dimensions $d=1,2,3$. Firstly, we show the global well-posedness, as well as the existence of global and exponential attractors with finite fractal dimensions for these problems. In the case of the exchange-dominated LLBar equation and the CH-AC equation without convection, an estimate for the rate of convergence of the solution to the corresponding stationary state is given. Finally, we show the existence of a robust family of exponential attractors when $d\leq 2$. As a corollary, exponential attractor of the LLBar equation is shown to converge to that of the Landau--Lifshitz--Bloch equation in the limit of vanishing exchange damping, while exponential attractor of the convective CH-AC equation is shown to converge to that of the convective Allen--Cahn equation in the limit of vanishing diffusion coefficient.