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1. Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

   https://doi.org/10.1111/sapm.12642  

   Alkın, Aykut; Mantzavinos, Dionyssios; Özsarı, Türker (January 2024, Studies in Applied Mathematics)

   Abstract We establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator. 

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2. Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential

   Duan, Renjun; Liu, Shuangqian; Sakamoto, Shota; Strain, Robert M. (June 2020, RIMS kokyuroku bessatsu)

   Takayoshi Ogawa; Keiichi Kato; Mishio Kawashita (Ed.)

   This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $$A(\Omega)$$, which can be expected to play a similar role to $$L^\infty$$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way. 

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3. A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases

   https://doi.org/10.4208/nmtma.2017.s12  

   Shu, Ruiwen; Hu, Jingwei; Jin, Shi (May 2017, Numerical Mathematics: Theory, Methods and Applications)

   Abstract We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel. 

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4. The inviscid limit for the 2D Navier-Stokes equations in bounded domains

   https://doi.org/10.3934/krm.2022004  

   Bardos, Claude W.; Nguyen, Trinh T.; Nguyen, Toan T.; Titi, Edriss S. (January 2022, Kinetic and Related Models)

   We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the standard wellposedness theory of Navier-Stokes equations in Sobolev spaces away from the boundary. 

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5. Traveling Wave Solutions to the One-Phase Muskat Problem: Existence and Stability

   https://doi.org/10.1007/s00205-023-01951-z  

   Nguyen, Huy Q; Tice, Ian (February 2024, Archive for Rational Mechanics and Analysis)

   We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem. 

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