Mathematische Nachrichten最新文献

The trace-dev-div inequality in $�H^s$ controls the trace in the norm of $�H^s$ by that of the deviatoric part plus the $�H^{�s�-�1�}$ norm of the divergence of a quadratic tensor field different from the constant unit matrix. This is well known for $��s�=�0�$ and established for orders $��0�\le �s�\le �1�$ and arbitrary space dimension in this paper. For mixed and least-squares finite element error analysis in linear elasticity, this inequality allows to establish robustness with respect to the Lamé parameter $�\lambda$.

In this paper, we investigate the two-dimensional (2D), two-field magnetohydrodynamics (MHD) equations in the presence of a shear flow, assuming positive plasma viscosity and resistivity. We establish the global-in-time existence and uniqueness of a strong solution for the 2D two-field MHD equations under shearing-periodic boundary conditions, as proposed by Hawley et al. Moreover, we establish the existence and uniqueness of a strong solution for the linear advection-diffusion equation under shearing-periodic boundary condition by employing uniformly local $�L^2$ spaces.

In this paper, we study the nonlinear dissipative Boussinesq equation in the whole space $�R^n$ with $�L^1$ integrable data. As our preparations, the optimal estimates as well as the optimal leading terms for the linearized model are derived by performing the Wentzel–Kramers–Brillouin (WKB) analysis and the Fourier analysis. Then, under some conditions on the power $�p$ of nonlinearity, we demonstrate global (in time) existence of small data Sobolev solutions with different regularities to the nonlinear model by applying some fractional-order interpolations, where the optimal growth ($��n�=�2�$) and decay ($��n�⩾�3�$) estimates of solutions for large time are given. Simultaneously, we get a new large time asymptotic profile of global (in time) solutions. These results imply some influence of dispersion and dissipation on qualitative properties of solution.

We study the existence, uniqueness and polynomial stability of forward asymptotically almost periodic (AAP-) mild solutions for the wave equation with a singular potential on the whole space $�R^n$ in a framework of weak-$�L^p$ spaces. First, we use a Yamazaki-type estimate for wave groups on Lorentz spaces to establish the global well-posedness of bounded mild solutions for the corresponding linear wave equations. Then, we provide a Massera-type principle which guarantees the existence of AAP-mild solutions for linear wave equations. Using the results of linear wave equations and fixed point arguments we establish the well-posedness of such solutions for semilinear wave equations. Finally, we obtain a polynomial stability for mild solutions by employing dispersive estimates.

It is known that there exist complex solvmanifolds $��(�\Gamma �\setminus �G�,�J�)�$ whose canonical bundle is trivialized by a holomorphic section that is not invariant under the action of $�G$. The main goal of this paper is to classify the six-dimensional Lie algebras corresponding to such complex solvmanifolds, thus extending the previous work of Fino, Otal, and Ugarte for the invariant case. To achieve this, we complete the classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures and identify among them, the ones admitting complex structures with Chern–Ricci flat metrics. Finally, we construct complex solvmanifolds with non-invariant holomorphic sections of their canonical bundle. In particular, we present an example of one such solvmanifold that is not biholomorphic to a complex solvmanifold with an invariant holomorphic section of its canonical bundle. Additionally, we discover a new six-dimensional solvable strongly unimodular Lie algebra equipped with a complex structure that has a nonzero holomorphic (3,0)-form.

In this paper, we consider the asymptotic behavior of solutions to the Cauchy problem for the three-dimensional model of compressible reactive fluid, which can be described by a compressible Navier–Stokes type system with potential external force. First, the existence of the stationary solution is shown in the case that the external force is small enough. Next, making use of the energy method, we prove that the stationary solution is time-asymptotically stable provided that the external force and the initial perturbation are sufficiently small. Finally, we obtain the time-decay rate of the solution toward the stationary solution by combining the $��L^p�-�L^q�$ estimates for the corresponding linear problem and the energy estimates for the nonlinear system.

We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order.

The setting that we introduce is very general and comprises, for instance, the sum of two fractional Laplacians, or of a fractional Laplacian and a Laplacian, as particular cases (the situation in which there are infinitely many operators, and even a continuous distribution of operators, can be considered as well).

New bits of functional analysis are introduced to deal with this problem. An eigenvalue analysis divides the existence theory into two streams, one related to a mountain pass method, the other to a linking technique.

The uniformization theory of Gromov hyperbolic spaces investigated by Bonk, Heinonen, and Koskela, generalizes the case where a classical Poincaré ball type model is used as the starting point. In this paper, we develop this approach in the case where the underlying domain is unbounded, corresponding to the classical Poincaré half-space model. More precisely, we study conformal densities via Busemann functions on Gromov hyperbolic spaces and prove that the deformed spaces are unbounded uniform spaces. Furthermore, we show that there is a one-to-one correspondence between the bilipschitz classes of proper geodesic Gromov hyperbolic spaces that are roughly starlike with respect to a point on the Gromov boundary and the quasisimilarity classes of unbounded locally compact uniform spaces. Our result can be understood as an unbounded counterpart of the main result of Bonk, Heinonen, and Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque. 270 (2001).