In this paper, we construct a new class of uniformly rotating vortex sheets arising as perturbations of the circular configuration within the Kelvin–Helmholtz model for two incompressible fluids of equal density, with surface tension effects taken into account. Despite the fundamental importance of the problem, few results are available on the long-time dynamics, due to the strong instability inherent in the Kelvin–Helmholtz mechanism. We carry out a detailed bifurcation analysis and construct families of nontrivial rotating solutions that bifurcate from the circular state when either the rotational speed or the surface tension coefficient is varied. The stationary branches obtained by varying the mean vorticity follow from the same analysis through the scaling symmetry of the equations.
In this paper, we investigate the uniform stability and optimal time decay of strong solutions to the incompressible kinetic-magnetohydrodynamic (kinetic-MHD) system in the whole space ℝ^3 . This model consists of a kinetic equation describing the evolution of energetic particles and the incompressible MHD equations governing the dynamics of the fluid and magnetic field, coupled through Lorentz forces. Under the assumption that the initial data are small perturbations near the spatially homogeneous equilibrium (M, 0, 0), we first establish the global existence of strong solutions in the L_v^2(H_x^2)× H_x^2× H_x^2 framework, without imposing any regularity assumption on the velocity derivatives of the kinetic perturbation f. Under an additional smallness assumption in L^1 , by employing the low-high frequency decomposition method and the time-weighted energy method, we address the difficulties arising from the velocity-weighted term and the loss of velocity derivatives in v× B·∇ _v f . As a result, we obtain the optimal time decay rates of all spatial derivatives up to order two, including the highest-order rate (1+t)^-7/4 . Here, a=∫ _ℝ^3√(M)f dv and b=∫ _ℝ^3v√(M)f dv denote the zeroth and first velocity moments of f, respectively. We further identify a magnetic-field-induced dissipation mechanism such that b and ∇ b decay one-half order faster than the corresponding orders of the full solution, while accelerated decay also holds for ∂ _t a and ∂ _t b . Furthermore, a refined difference-energy method yields the uniform stability of strong solutions. In the periodic domain 𝕋^3 , we show that f and B decay exponentially, whereas the fluid velocity u remains uniformly bounded, which differs from kinetic-fluid systems with drag-force coupling. To the best of our knowledge, this is the first result concerning the uniform stability and optimal time decay of strong solutions for this kinetic-MHD model.
Based on the work of Chae [4], we establish two refined Liouville-type theorems for the three-dimensional stationary Navier–Stokes equations. The main idea of this paper is to construct divergent series via the summation of integrals over mutually disjoint level-set shells, which leads to contradictions with the finite Dirichlet energy of velocity and the finite L^3/2 norm of the gradient of the total head pressure, respectively. This approach allows for a logarithmic growth factor to be included in the polynomial decay requirements previously prescribed in Chae’s work [4].
This paper studies the initial boundary value problems for the 2D MHD equations with vertical dissipation and horizontal magneto damping in a strip domain. The proof is nontrivial and relies on a sophisticated analytical framework involving anisotropic norms, time-derivative estimates, and time-weighted energy functionals. To address the challenges posed by weak dissipation and boundary effects, we develop a set of anisotropic inequalities and establish key auxiliary estimates. For sufficiently small initial data, a careful analysis of the nonlinear terms then leads to the global well-posedness of the system in the Sobolev setting H^3 . Furthermore, the H^2 -norm of the velocity field and the magnetic field are shown to decay exponentially in time.
This paper establishes a priori estimates for the free boundary problem of two-dimensional ideal incompressible magnetohydrodynamics (MHD) involving a plasma-vacuum interface. The system consists of a plasma region governed by the ideal MHD equations and a vacuum region described by pre-Maxwell dynamics, separated by a freely evolving interface where total pressure continuity and magnetic field tangency conditions hold. By adopting a geometric Lagrangian framework, we reformulate the free boundary problem into a fixed domain using trajectory maps and fictitious velocity extensions. Our main contribution lies in deriving higher-order energy norms that combine boundary geometry and interior dynamics, enabling control over the solution’s regularity. For the vacuum magnetic field, we prove that its covariant derivatives in L^2 norms are bounded solely by initial data and curvature parameters, leveraging geometric trace inequalities under bounded curvature constraints. Additionally, we establish evolution equations for the electric field in vacuum and demonstrate energy exchanges between plasma and vacuum regions through pressure balance. The results provide a crucial step towards proving well-posedness for this challenging interface problem.
In this paper, we consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain involving a free moving surface boundary with surface tension. We derive the a priori estimate for solutions under minimal regularity assumptions on the initial data in Lagrangian coordinates. To the best of our knowledge, this seems to be the first attempt to establish the result of low regularity solution for incompressible free-boundary MHD equations with surface tension. Such result extends the works of Luo and Zhang [SIAM, 53 (2021), pp. 2595-2630] and Disconzi, Kukavica and Tuffaha [SIAM, 51 (2019), pp. 3982-4022].
We study the Cauchy problem for a nonlocal evolution model arising in equatorial ocean flows. We prove global existence of solutions for small initial data in critical Besov spaces and show that these solutions become instantly spatially analytic. Our approach introduces exponentially weighted unknowns to capture the interplay between the dissipative linear operator and the nonlocal Hilbert transform, allowing precise control of the bilinear nonlinearity. This provides a unified framework to extend small-data global existence and analyticity results from classical fluid equations to a nonlocal dispersive-dissipative setting.
In this article, we study a thermohaline convection in horizontal fluid layer such that the gravity is considered. The fluid is heated from above non-uniformly. An existence theorem of stationary solutions is provided by a fixed point argument. Some interesting flow patterns are demonstrated by numerical computations.
In this paper, we consider the quantum magnetohydrodynamic model for quantum plasmas. We first derive uniform estimates for the global smooth solutions in terms of the quantum coefficient ħ and the Hall coefficient ϵ . Then we establish the existence of global solutions and derive optimal convergence rates using the energy method. Next, applying the Lions-Aubin lemma, we prove that the unique smooth solution of the three-dimensional Hall-quantum-magnetohydrodynamic system converges globally in time to the smooth solution of the three-dimensional Navier-Stokes system as ħ and ϵ tend to zero. Furthermore, we provide the convergence rate estimates for any given positive time.
There have already been some results concerning on the global existence of solutions to the Cauchy problem of the planar compressible magnetohydrodynamic equations with large initial data. However, few results have focused on its large-time behavior. The main purpose of this paper is to study the large-time behavior of global solutions to the Cauchy problem of the above system. The viscosity can be a positive constant or density-dependent. The key point in our analysis is to derive the uniform-in-time positive lower and upper bounds on the specific volume and the absolute temperature. Moreover, the interaction between the hydrodynamics and magnetodynamic effects has also been dealt with properly.
We consider the axisymmetric Navier-Stokes equations in a finite cylinder Ω⊂ℝ^3 . We assume that v_r , v_φ , ω _φ vanish on the lateral part of boundary ∂Ω of the cylinder, and that v_z , ω _φ , ∂ _zv_φ vanish on the top and bottom parts of the boundary ∂Ω , where we used standard cylindrical coordinates, and we denoted by ω =curlv the vorticity field. We use H^3 Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Finally, using the estimate ‖ v_φ‖ _L_q(0,T;L_p(Ω ))≤ A, where A is a given number and 3/p + 2/q < 1 , q<∞ we prove the existence of global regular axially-symmetric solutions.
In a three-dimensional bounded domain Ω we consider the compressible Navier-Stokes equations for a barotropic fluid with general non-linear density dependent viscosities and no-slip boundary conditions. A nonlinear drag term is added to the momentum equation. We establish two conditional Kato-type criteria for the convergence of the weak solutions to such a system towards the strong solution of the compressible Euler system when the viscosity coefficient and the drag term parameter tend to zero.
This article addresses the existence of solutions for partial differential equation (PDE) models describing granular flows. We emphasize the essential role of flow dilatation, coupled with complex rheology, in ensuring both stability and the existence of dissipative energy. A central focus of the paper is to understand how this energy, arising from strongly nonlinear and singular terms, contributes to the existence of weak solutions. We first establish an existence result for a model that reflects some mathematical difficulties of the complete system. In particular, the dilatancy law describes local volume changes in terms of the velocity divergence, which depends on the shear rate and the square root of the pressure, reflecting a balance between these two quantities. While the model rigorously studied in this article does not address all the difficulties of the full physical model - specifically, the variable volume fraction case is handled here solely through regularisation - this work represents a significant step forward in the mathematical analysis of models for such complex flows.
In this paper, we investigate wave breaking for the Fornberg-Whitham-Degasperis-Procesi equation which can be viewed as a special shallow water wave equation. Without any conservation laws, we give sufficient conditions on the initial data to lead to wave breaking.
We are concerned with the estimate of singular set of weak solution to Magneto-hydrodynamical (MHD) equations. First, we show that if the pressure P associated to a Leray-Hopf weak solution (u, b) satisfies some additional assumptions, then there is a reduction in the Hausdorff dimension of singular set at a first potential blow up time. Next, instead of imposing the assumption on the pressure, we prove that if the initial data satisfies an extra assumption, then (u, b) possess finite number singular points at first potential blow up time.
In this paper, we consider non-uniqueness and finite time blowup of the BV-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the p-system. The recent Bressan-De Lellis result [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] shows that whenever a BV solution exists, with finite (but possibly very large total variation), it is unique if each shock verifies the Liu E-condition. We show non-uniqueness of solutions by convex integration. The solutions we construct are Liu-admissible for a trivial reason: there are no shocks, so the Liu E-condition is vacuously satisfied. But our construction shows exactly that there is an issue, a qualitative difference between small data solutions (for example in the BV class) and large data solutions, where this exact type of phenomenon might occur. Our result can be interpreted as a cautionary example to show that the Liu E-condition is satisfactory for small oscillations but not for large oscillations where typically these type of constructions appear. In particular, we present Riemann initial data which admits infinitely many bounded solutions, each of which experience, not just finite time, but in fact instantaneous blowup of the BV norm. The Riemann initial data is allowed to come from an open set in state space. Our method provably does not admit the natural strictly convex entropy. The proof of our theorem is computer-assisted. Our code is available on the GitHub.
In this paper, we study the initial-boundary value problem for the Stokes system in the three-dimensional infinite layer domain Ω = (0,π/2) ×ℝ^2 , subject to upper stress-free and lower slip-type boundary conditions. Using the Fourier method, we derive an integral formulation for regular solutions. Based on this formula, we demonstrate the solvability of the problem in the L^2 -based Sobolev space H^2(Ω ) .
Let us consider the spatial pointwise behavior of time-periodic solutions to the Navier-Stokes equation in the exterior of a rigid body, moving by time-periodic motion. For the translational and angular velocity of the body, assuming besides smallness and regularity, either of the following conditions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. If time average over a period of translational velocity, λ (say), is non-zero (resp. zero), we then show that gradient of the velocity of the fluid decays like the one of the gradient of the Oseen fundamental solution (resp. decays at the rate O(|x|^-2) ). As applications, in the case λ =0 , we show the attainability of the time-periodic solution for small data. In the case λ 0 , the stability/attainability of the time-periodic solution with sharp decay properties are also deduced.
This paper is concerned with the explicit solutions for a class of geophysical gyre models. Under the relevant physical conditions, the current model can be regarded as a governing equation of the Antarctic Circumpolar Current (ACC). For the linear oceanic vorticity case, we derive some explicit two-dimensional solutions which depend on solving the associated Fuchs type equation, Hypergeometric equation, and Legendre's differential equation, respectively. These explicit solutions provide a framework for analyzing the influence of various parameters on the flow field and offer valuable insights into the characteristics of the ACC. Our simulations under constant vorticity capture the trend of the ACC's velocity increasing with latitude. Finally, for the nonlinear vorticity case, we establish the existence of nontrivial solutions to this equation by introducing energy functionals and imposing appropriate assumptions.
