For the three dimensional linearized MHD boundary layer system around a shear flow, we establish a well-posedness result with Sobolev regularity in one tangential direction under the non-degeneracy condition in that direction. The proof is based on an observation of a new cancellation mechanism that is different from the two space dimensional case. Precisely, the new cancellation relies on the evolution of the magnetic field orthogonal to the boundary instead of the stream function in two space dimension. Even though this kind of cancellation can help to lower the regularity requirement in only one tangential direction while analyticity is still needed in the other tangential direction, we expect that this can be viewed as one step further to study the challenging problem on the well-posedness theory in Sobolev space of the three dimensional MHD boundary layer system under some suitable structural assumption.

We consider the compressible Navier-Stokes equations for a reacting ideal polytropic gas when the coefficients of viscosity, thermal conductivity, and species diffusion are general smooth functions of temperature. The choice of temperature-dependent transport coefficients is motivated by the kinetic theory and experimental results. We establish the existence, uniqueness, and time-asymptotic behavior of global solutions for one-dimensional, spherically symmetric, or cylindrically symmetric flows under certain assumptions on the $H^2$ norm of the initial data. This is a Nishida-Smoller type global solvability result, since the initial perturbations can be large if the adiabatic exponent is close to 1.

We analyze a boundary value problem for the flow of a relativistic fluid confined in a bounded domain. An important point is that we base our problem on the Lagrange coordinates of the fluid which we define here, for the first time in the (1+1)-dimensional case. The Lagrange coordinates for the 1+3 relativistic flow was established in [Dias and Frid, Comm. Math. Anal. Appl. 2(3) (2023)].

We survey recent developments on the analysis of Gauss-Codazzi-Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions – in line with the pioneering works [Chen et al., Proc. Amer. Math. Soc. 138 (2010), Commun. Math. Phys. 294 (2010)] – on the weak continuity of Gauss-Codazzi-Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb-Uhlenbeck gauges.

In this article, we study the hypersonic limit problem to the 1-D isentropic Euler equations. For uniformly bounded density and velocity, it can be formulated as the behavior of solution as $\gamma→1.$ First we study and clarify the mechanism of singularity formation for two case: Only derivatives blow up when $\gamma>1,$ the derivatives blow up with mass concentrates when $\gamma=1.$ Then we showed as $\gamma→1,$ the classic solutions of the isentropic Euler equations converge to the solutions of the pressureless Euler equations. We proved that $u$ converges in $C^1$ and $ρ$ converges in $C^0.$ By a level set argument, the convergence rate is proved to be $\sqrt{\gamma-1}$ on any fixed level set. Furthermore, we show that the time that singularity forms for $\gamma>1$ converges to the time of singularity forms for $\gamma=1.$

In this paper, we present a new framework for the global well-posedness and large-time behavior of a two-phase flow system, which consists of the pressureless Euler equations and incompressible Navier-Stokes equations coupled through the drag force. To overcome the difficulties arising from the absence of the pressure term in the Euler equations, we establish the time decay estimates of the high-order derivative of the velocity to obtain uniform estimates of the fluid density. The upper bound decay rates are obtained by designing a new functional and the lower bound decay rates are achieved by selecting specific initial data. Moreover, the upper bound decay rates are the same order as the lower one. Therefore, the time decay rates are optimal. When the fluid density in the pressureless Euler flow vanishes, the system is reduced into an incompressible Navier-Stokes flow. In this case, our works coincide with the classical results by Schonbek [ J. Amer. Math. Soc. 4 (1991)], which can be regarded as a generalization from a single fluid model to the two-phase fluid one.