The fragmentation equation with size diffusion: Well posedness and long-term behaviour

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2<p<∞ and N<q<∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

The Super-Diffusive Singular Perturbation Problem

In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] , 1 < α < 2 , ϵ > 0 , for the parabolic equation ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem ( P ϵ ) has a unique solution u ϵ ( t ) for each small ϵ > 0 . Moreover u ϵ ( t ) converges to u 0 ( t ) as ϵ → 0 + , the unique solution of equation ( P ) .

Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder continuity in $x$, but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (1) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

Operators with Gaussian kernel bounds on mixed Morrey spaces

Let L be an analytic semigroup on L2(Rn) with Gaussian kernel bound, and let L-?/2 be the fractional operator associated to L for 0 < ? < n. In this paper, we prove some boundedness properties for the commutator [b,L-?/2] on Mixed Morrey spaces Lq,? (0,T,Lp,?(Rn)), when b belongs to BMO(Rn) or to suitable homogeneous Lipschitz spaces.