Article
Computer Science, Interdisciplinary Applications
Markus Kivioja, Sanna Monkola, Tuomo Rossi
Summary: This paper presents a reliable numerical method and efficient GPU-accelerated implementation for the time integration of the three-dimensional Gross-Pitaevskii equation. The method utilizes discrete exterior calculus and offers more versatile spatial discretization compared to traditional methods. The implementation achieves significant speedups on the GPU and is further parallelized to multiple GPUs.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
M. E. N. G. Yuan
Summary: This article focuses on the initial-boundary value problems of 3-D quasilinear wave equations outside compact convex obstacles with Neumann boundary conditions. It establishes the almost global existence of smooth small-amplitude solutions to these problems when the surfaces of the obstacles are smooth and the quadratic nonlinearities do not fulfill the null condition. The lower bound of the lifespan is proven to be optimal, as shown in the 3-D boundaryless case.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Binhua Feng, Leijin Cao, Jiayin Liu
Summary: This paper investigates the existence of stable standing waves for the Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation with a partial harmonic confinement. The results show that stable standing waves exist under certain conditions, complementing previous research on the topic.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Linjie Song
Summary: By using bifurcation type arguments, we establish threshold results for the existence, nonexistence, and multiplicity of positive solutions with prescribed L-2 norm for a semi-linear elliptic equation in bounded domains. We do not assume f to be autonomous. Furthermore, we provide a lower bound for the threshold. For almost every L-2 mass in the existence range, there exist one orbitally stable standing wave and one unstable standing wave associated with these positive solutions. We also study the existence of prescribed norm solutions in exterior domains and orbitally unstable standing waves for almost every L-2 mass in the existence range.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Dana Mendelson, Andrea R. Nahmod, Natasa Pavlovic, Matthew Rosenzweig, Gigliola Staffilani
Summary: This paper studies the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. A series of observables are established such that the corresponding trace functionals, referred to as energies, commute with respect to the weak Lie-Poisson structure defined by the authors in [57]. The Hamiltonian equation associated with the third energy functional precisely corresponds to the GP hierarchy, while the equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schrodinger hierarchy.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Motohiro Sobajima
Summary: This paper investigates the existence and non-existence of global-in-time solutions of a weakly coupled parabolic system, providing regions for solutions in both whole space and exterior domains, highlighting critical cases and differences between them due to the behavior of linear two-dimensional heat semigroup.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Claudianor O. Alves, Vincenzo Ambrosio, Cesar E. Torres Ledesma
Summary: This paper examines the existence of solutions for a class of magnetic semilinear Schrodinger equations under certain conditions.
MILAN JOURNAL OF MATHEMATICS
(2021)
Article
Mathematics, Applied
Yongming Luo, Athanasios Stylianou
Summary: This paper establishes the existence and qualitative properties of ground state solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation. By utilizing a mountain pass argument on spheres and constructing appropriately localized Palais-Smale sequences, the existence of real positive ground states as saddle points is proven. Additionally, a nonlocal Pohozaev identity with no rest term is also demonstrated, which is a crucial part of the analysis.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Wei Xu
Summary: This paper establishes the global existence of classical solutions to systems of nonlinear wave equations with multiple speeds outside star-shaped regions satisfying time-independent inhomogeneous boundary conditions, provided that nonlinear terms obey the null condition. We first prove the existence and uniqueness of stationary solutions. Then we use the space-time estimates for perturbed wave equations in Metcalfe and Sogge (2007) to show that solutions of systems converge to stationary solutions as time goes to infinity.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Physics, Mathematical
Yunkun Chen, Yi Peng, Xue Wang
Summary: This paper investigates the full compressible magnetohydrodynamic system in three-dimensional exterior domains and establishes the global existence and uniqueness of strong solutions for the initial-boundary-value problem under specific boundary conditions. The large-time behavior of the strong solutions is also shown.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Xueying Yu
Summary: The Cauchy initial value problem for the defocusing quintic nonlinear Schrodinger equation in two dimensions with general data in the critical space (H)over dot(1/2)(R-2) is considered. It is shown that if a solution remains bounded in (H)over dot(1/2)(R-2) in its maximal interval of existence, then the interval is infinite and the solution scatters.
Article
Mathematics, Applied
Mihaela Ifrim, Daniel Tataru
Summary: This article introduces a new nonperturbative method to prove global well-posedness and scattering for one-dimensional NLS problems with cubic nonlinearity. The method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago.
FORUM OF MATHEMATICS PI
(2023)
Article
Mathematics
Keiichi Watanabe
Summary: This paper develops Lp-Lq decay estimates of the gradient of the Stokes semigroup (T (t))t >= 0 generated by the negative of the Stokes operator in exterior Lipschitz domains Q subset of Rn, n >= 3. The Lp-Lq estimates of backward difference T (t) with optimal rates are proved if p and q satisfy |1/p - 1/2| < 1/(2n) + 6, |1/ q - 1/2| < 1/(2n) + c, and p <= q <= n with some c > 0. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier-Stokes equations in the critical space L infinity(0, infinity; L3 sigma (Q)) provided that the initial velocity is small in the L3-norm.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Hui Zhang, Junxiang Xu
Summary: This paper investigates the existence of ground states for the singularly perturbed Gross-Pitaevskii equation with small ε, and describes the concentration phenomena of ground states as ε approaches 0. The relationship between the number of positive solutions and the profile of the potential V is also explored.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2021)
Article
Mathematics, Applied
Jiayin Liu, Zhiqian He, Binhua Feng
Summary: This paper investigates the existence and stability of stable standing waves for the inhomogeneous Gross-Pitaevskii equation with partial confinement in both the L-2-subcritical and L-2-supercritical cases. The results complement previous studies on standing waves with complete confinement and contribute to the existing research in this field.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Miguel Brozos-Vazquez, Diego Mojon-Alvarez
Summary: We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped product in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2024)
Article
Mathematics, Applied
Domenec Ruiz-Balet, Enrique Zuazua
Summary: Inspired by normalising flows, we analyze the bilinear control of neural transport equations using time-dependent velocity fields constrained by a simple neural network assumption. We prove the L1 approximate controllability property, showing that any probability density can be driven arbitrarily close to any other one within any given time horizon. The control vector fields are explicitly and recursively constructed, providing quantitative estimates of their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2024)
