In this paper, we study the problem of energy conservation for compressible magnetohydrodynamic(MHD) system in (0,T)×T3. We obtain sufficient regularity conditions of weak solutions for the MHD system to satisfy the energy conservation. This work is inspired by Feireisl et al. [1] and Wiedemann et al. [2], which study the energy conservation problem of compressible isentropic Euler equations and Navier-Stokes equations. Compared with [1,2], we consider the MHD equations with a general viscosity coefficient, which makes the analysis more complicated.
We study a mathematical model describing the dynamic behavior of a viscoelastic body in frictional contact with a rigid foundation. The contact is governed by simplified Coulomb’s friction law, and the material exhibits a nonlinear constitutive law incorporating long-term memory effects. We present the model, specify the assumptions on the data, and derive a variational formulation in terms of the displacement field. We first establish a local existence and uniqueness result, then the global existence result is proved via energy methods. Finally, we prove the exponential stability of energy solutions using the multiplier technique.
This paper focuses on a system of the two-dimensional magnetohydrodynamic equations with horizontal fractional dissipation in the domain Ω=T×R with T=[0,1] being a periodic box. We study the stability problem on perturbations near any fixed magnetic field A=(A1,A2) where A1,A2∈R. Global stability for this system was proven by Feng et al. [1] in the Sobolev setting H3(Ω); however, it seems that one can not establish global stability for the lower regularity solutions to this system by using their method, such as in the Sobolev setting H2(Ω). The present work resolves this problem and the global stability for this system in H2(Ω) is obtained by utilizing the anisotropic interpolation inequalities and the general strong Poincaré type inequalities, which relaxes the result in [1]. In addition, we also show that the oscillation part of the solution converges to zero in H1(Ω) as t → ∞.
A one-dimensional Keller-Segel equation with a piecewise-defined reaction term depending on the density gradient gives rise to a nonsmooth fast-slow system. Using Filippov convexification and geometric singular perturbation theory, we prove the existence of sliding traveling wave solutions for sufficiently small ε > 0 and characterize the corresponding sliding region. In the singular limit ε=0, we construct a singular sliding heteroclinic orbit on the critical manifold and show that it persists for sufficiently small ε > 0 under appropriate conditions on the wave speed and model parameters. The family of perturbed orbits converges to the singular orbit in the Hausdorff distance as ε → 0. These results provide a rigorous construction of sliding traveling waves for a chemotaxis model with gradient-dependent switching.
This paper is devoted to the study of a new partial differential system involving an evolutionary variational inequality, a history-dependent hemivariational inequality and a partial differential equation in Banach spaces. Under mild conditions, by applying Banach’s fixed point theorem, we prove a unique solvability result for the problem. Furthermore, the main results are applied to obtain the unique solvability of a long-memory viscoelastic frictional contact problem that incorporates wear, damage, and a curved contact surface.
