Journal of Differential Equations最新文献

We study the boundary weighted regularity of weak solutions u to a s-fractional p-Laplacian equation in a bounded $C^{1,1}$ domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with $s\in (0,1)$. We prove optimal up-to-the-boundary regularity of u, which is $C^s\left(\bar{\Omega }\right)$ for any $p>1$ and, in the singular case $p\in (1,2)$, that $u/d_{\Omega }^s$ has a Hölder continuous extension to the closure of Ω, $d_{\Omega }\left(x\right)$ meaning the distance of x from the complement of Ω. This last result is the singular counterpart of the one in [30], where the degenerate case $p⩾2$ is considered.

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$ with $p\in (1,\infty$). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.

A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line $R$. Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values $V_0\left(x_0\right)$ and $V_0^{\prime }\left(x_0\right)$ at a single local point $x_0\in R$.

This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp $L^{\infty }$ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups $G$ with the homogeneous dimension N. We consider the nonlinear function behaves like $|u{|}^{\alpha }$ or $|u{|}^{\alpha -1}u$ $(\alpha >1)$ and the initial data $u_0$ belongs to the Sobolev spaces $L_s^p\left(G\right)$ for $1<p<\infty$ and $0<s<N/p$. Since stratified Lie groups $G$ include the Euclidean space $R^n$ as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on $R^n$ to $G$. It should be noted that our proof is very different from it given by Ribaud on $R^n$. We adopt the generalized fractional chain rule on $G$ to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on $R^n$. By using the generalized fractional chain rule on $G$, we can avoid the discussion of Fourier analysis on $G$ and make the proof more simple.

In this paper, we consider the homogenization of evolutionary incompressible purely viscous non-Newtonian flows of Carreau-Yasuda type in porous media with small perforation parameter $0<\epsilon \ll 1$, where the small holes are periodically distributed. Darcy's law is recovered in the homogenization limit. Applying Poincaré type inequality in porous media allows us to derive the uniform estimates on the velocity field, the gradient of which is small of size ε in $L^2$ space. This indicates the nonlinear part in the viscosity coefficient does not contribute in the limit and a linear model (Darcy's law) is obtained. The estimates of the pressure rely on a proper extension from the perforated domain to the homogeneous non-perforated domain. By integrating the equations in time variable such that each term in the resulting equations has certain continuity in time, we can establish the extension of the pressure by applying the dual formula with the restriction operator.

We consider the Einstein-Boltzmann system for massless particles in the Bianchi I space-time with scattering cross-sections in a certain range of soft potentials. We assume that the space-time has an initial conformal gauge singularity and show that the initial value problem is well posed with data given at the singularity. This is understood by considering conformally rescaled equations. The Einstein equations become a system of singular ordinary differential equations, for which we establish an existence theorem which requires several differentiability and eigenvalue conditions on the coefficient functions together with the Fuchsian conditions. The Boltzmann equation is regularized by a suitable choice of time coordinate, but still has singularities in momentum variables. This is resolved by considering singular weights, and the existence is obtained by exploiting singular moment estimates.

We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. Explicit solutions to such a water wave problem are static configurations where the surface is an unduloid, that is, a periodic surface of revolution with constant mean curvature. We prove that to any such configuration there connects a global continuum of non-static solutions by means of a global implicit function theorem. To prove this, the key is strict monotonicity of a certain function describing the mean curvature of an unduloid and involving complete elliptic integrals. From this point of view, this paper is an interesting interplay between water waves, geometry, and properties of elliptic integrals.

In this article, we develop a linear theory to deal with the time periodic problem for the Navier-Stokes equations on unbounded domains with moving boundary. Compared to the case of bounded domains the underlying modified time-dependent Stokes operators are no longer invertible, thus leading to a more sophisticated construction of the evolution operator. Moreover, Sobolev embeddings on $L^q$ spaces imply restrictions on q depending on geometric properties of the domain. The theory is focusing on the half space case, the construction and local-in-time estimates of the evolution operator and its adjoint in view of time periodic solutions.

In this paper, we are concerned with the following Schrödinger system$\{\begin{array}{cc}-\Delta u+P\left(x\right)u={\mu }_1{u}^3+{\beta }_{1}u{v}^2+{\beta }_{2}u{w}^2,& x\in R^3,\\ -\Delta v+Q\left(x\right)v={\mu }_2{v}^3+{\beta }_{1}v{u}^2+{\beta }_{3}v{w}^2,& x\in R^3,\\ -\Delta w+\lambda w={\mu }_3{w}^3+{\beta }_{2}w{u}^2+{\beta }_{3}w{v}^2,& x\in R^3,\end{array}$ where $\lambda >0$ is a positive constant, $P\left(x\right),Q\left(x\right)$ are continuous positive radial potentials, ${\mu }_i(i=1,2,3)>0$ and ${\beta }_i(i=1,2,3)\in R$ are coupling constants. We mainly investigate the effect of the potentials and the nonlinear coupling on the structure of solutions. Applying the Lyapunov-Schmidt redu