Mathematische Nachrichten最新文献

In this paper, we establish a weighted Kato–Ponce inequality for $�m$ factors in the endpoint case. Furthermore, we extend the validity of the Kato–Ponce inequality from the class of Schwartz functions to the broader class of functions living in a (weighted) fractional Sobolev space.

We study connections between the $�W_p^1$-differentiability and the $�L_p$-differentiability of Sobolev functions. We prove that $�W_p^1$-differentiability implies the $�L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $�W_p^1$-differentiability of Sobolev functions $�{cap}_p$-almost everywhere.

In this paper, we demonstrate the Wong–Zakai approximation results for two dimensional stochastic Cahn–Hilliard–Navier–Stokes model. The model consists of a Navier–Stokes system coupled with convective Cahn–Hilliard equations. It describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids under the influence of multiplicative noise. Our main result describes the support of the distribution of solutions. As in [2], both inclusions are proved by means of a general Wong–Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite-dimensional approximation of this process. Note that the coupling between the Navier–Stokes system and the Cahn–Hilliard equations makes the analysis more involved.

We investigate fractional operators with homogeneous kernel in Morrey spaces. In particular, we prove that fractional integral operators and fractional maximal operators with homogeneous kernel are bounded from the Calderón product of Morrey spaces to certain Morrey spaces. Our results can be seen as a generalization of a recent result on the relation between the boundedness of (classical) fractional operators and interpolation of Morrey spaces. What is new about this paper is not only the passage from the classical fractional integral operators to the rough integral operators. Even the case of fractional integral operators, handled in earlier papers, is significantly simplified.

We consider the Cauchy problem in $�R^n$ for the wave equation with a higher derivative term. We derive sharp growth estimates of the $�L^2$-norm of the solution itself for the case of $��n�=�1�$ and $��n�=�2�$. By imposing the weighted $�L^1$-initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of $��n�\ge �3�$, we observe that the $�L^2$-growth behavior of the solution never occurs in the $��(�L^2�\cap �L^1�)�$-framework of the initial data.

We study germs of real-analytic Levi-flat hypersurfaces with singularities on a boundary manifold. We prove the existence of a normal form for a real-analytic Levi-flat hypersurface which is defined by the vanishing of the real part of $�B_k$, $�C_k$, and $�F_4$ singularities. Finally, we prove a version of Morse–Vey's lemma for a real-analytic Levi-flat hypersurface with singularities on the boundary.

Control properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in $��(�0�,�+�\infty �)�$. Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.

We study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted $�N$-Laplace operator and without the mass term, and a higher-order fractional Poisson equation. Since the system is set in $�R^N$, the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign-changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case $��N�=�2�$.