Abstract We consider the three-dimensional incompressible magnetohydrodynamic system. Any weak solution satisfying a strong energy inequality is L 2 -asymptotically stable around a Landau solution. Under an additional integrability assumption on the initial perturbation, we also obtain an explicit algebraic decay rate for the L 2 -norm of the velocity and magnetic perturbations.
Abstract In this paper, we present a rigorous investigation of the asymptotic behaviour of the soliton gas in the modified short pulse (mSP) equation, where two generalized forms of the reflection coefficient on the interval are considered. The mSP-soliton gas is described through a Riemann–Hilbert (RH) problem, and further its asymptotic behaviour is analysed using the Deift–Zhou steepest descent method to the RH problem. One key step is the construction of the g -function, which differs significantly from the standard construction in the KdV equation since the mSP equation is a negative flow of the Wadati–Konno–Ichikawa system. To handle the singularity at the origin, we introduce a novel piecewise construction of the g -function. A key part of the analysis to estimate the error term involves constructing local parametrices near the endpoints and around the singularity. For the endpoints, the Airy parametrix and the first type of modified Bessel parametrix are employed, while for the singularity, we use the second type of modified Bessel parametrix and confluent hypergeometric parametrix.
Abstract This paper focuses on the initial-boundary value problem for the three-dimensional incompressible magnetohydrodynamic (MHD) equations with horizontal dissipation in the half space Ω := R + 3 under the slip boundary conditions. The goal here is to assess the stability and long-time behavior of perturbations near a background magnetic field B ¯ = ( 1 , 0 , 0 ) . By utilizing the structure of the system and boundary conditions to overcome difficulties arising from boundary integral terms, we prove the global stability in the Sobolev space H 3 ( R + 3 ) . After taking the Fourier cosine transforms in x h = ( x 1 , x 2 ) and the Fourier sine transforms in x 3 , and by the means of bootstrapping argument, we obtain the decay rates for the global solution itself as well as its derivatives.
Abstract This paper investigates the decay properties of solutions to a class of fractional transport-diffusion equations. For shear flow without critical points, we establish enhanced dissipation for all diffusion exponents s ∈ ( 0 , 1 ] . For shear flow with finitely many nondegenerate critical points, enhanced dissipation is obtained for all s ∈ ( 0 , 1 ] when n ⩾ 3 , and for s ∈ ( 1 / 3 , 1 ] when n = 1 , 2 . In the remaining low-dimensional regime n = 1 , 2 and 0 < s ⩽ 1 / 3 , we obtain decay on the diffusion time scale. To establish our results, we use the time-weighted hypocoercive energy method, which provides precise control while significantly reducing the regularity requirements on the initial data. Our analysis also employs interpolation inequalities and Hardy-type inequalities to derive more refined energy estimates.
Abstract This paper focuses on a family of generalized Laurent biorthogonal polynomials (LBOPs) together with the corresponding fully discrete integrable lattice, and their applications in matrix eigenvalue computations. Specifically, generalized LBOPs are investigated, including explicit expressions, two discrete spectral transformations, the recurrence relation, and spectral analysis. Furthermore, a generalized fully discrete relativistic Toda lattice equation is derived as the compatibility condition of these two discrete spectral transformations. It is shown that the discrete equation on a finite lattice is closely related to a generalized matrix eigenvalue problem and exhibits N conserved quantities associated with the generalized eigenvalues. Finally, the finite lattice equation is adapted as a matrix eigenvalue algorithm, for which convergence analysis and numerical examples are presented. It turns out that under specific initial conditions, the iterative process preserves the sign of the variables without encountering any singularities, and the corresponding generalized eigenvalues are real, simple, and positive. The comparison with the eig function (based on QZ algorithm) in MATLAB indicates that our algorithm can efficiently compute all eigenvalues and eigenvectors for a specific class of matrix pairs.
