Maximal regularity and a singular limit problem for the Patlak–Keller–Segel system in the scaling critical space involving BMO

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter $$\tau \rightarrow \infty $$ τ → ∞ . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces $$\dot{B}^s_{q,\sigma }({\mathbb {R}}^n)$$ B ˙ q , σ s ( R n ) and $$\dot{F}^s_{q,\sigma }({\mathbb {R}}^n)$$ F ˙ q , σ s ( R n ) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

Singular limit of Lam\’{e} equations

In this paper, we study the asymptotic behavior of the solution to the Lam\’{e} equations with a parameter $\var$. We prove that the solution will converge to the solution of a Maxwell type equations as $\var\rightarrow0$; Meanwhile we will show that the solution converges to the solution of a Stokes type equations as $\var\rightarrow\infty$.

A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy

An explicit representation of the Gamma limit of a single-well Modica–Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L 1 L^{1} -convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi–Warren–Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness are useful to characterize the limit of minimizers of the Kobayashi–Warren–Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problems is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

The limit problem of the Patlak-Keller-Segel-Stokes system in scalling critical space

In this paper, we consider a singular limit problem of Cauchy problem for Patlak-Keller-Segel equation coupled with Stokes equation in scalling critical space. Precisely, by taking advantage of a coupling structure of equations and using a scale decomposition technique, it is shown that when the relaxation time parameter ε →∞, a solution of Patlak-Keller-Segel system coupled with nonstationary Stokes equation converges to that of Patlak-Keller-Segel system coupled with stationary Stokes equation in the critical Fourier-Besov space under certain conditions.

Nonlocal characterizations of variable exponent Sobolev spaces

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.