2,072 research outputs found
Forward Discretely Self-Similar Solutions of the MHD Equations and the Viscoelastic Navier-Stokes Equations with Damping
In this paper, we prove the existence of forward discretely self-similar
solutions to the MHD equations and the viscoelastic Navier-Stokes equations
with damping with large weak initial data. The same proving techniques
are also applied to construct self-similar solutions to the MHD equations and
the viscoelastic Navier-Stokes equations with damping with large weak
initial data. This approach is based on [Z. Bradshaw and T.-P. Tsai, Ann. Henri
Poincar'{e}, vol. 18, no. 3, 1095-1119, 2017]
Tunable current circulation in triangular quantum-dot metastructures
Advances in fabrication and control of quantum dots allow the realization of
metastructures that may exhibit novel electrical transport phenomena. Here, we
investigate the electrical current passing through one such metastructure, a
system composed of quantum dots placed at the vertices of a triangle. The wave
natural of quantum particles leads to internal current circulation within the
metastructure in the absence of any external magnetic field. We uncover the
relation between its steady-state total current and the internal circulation.
By calculating the electronic correlations in quantum transport exactly, we
present phase diagrams showing where different types of current circulation can
be found as a function of the correlation strength and the coupling between the
quantum dots. Finally, we show that the regimes of current circulation can be
further enhanced or reduced depending on the local spatial distribution of the
interactions, suggesting a single-particle scattering mechanism is at play even
in the strongly-correlated regime. We suggest experimental realizations of
actual quantum-dot metastructures where our predictions can be directly tested.Comment: 5 pages, 4 figures, the Supplemental Information is attached at the
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Global Existence and Aggregation of Chemotaxis-fluid Systems in dimension two
To describe the cellular self-aggregation phenomenon, some strongly coupled
PDEs named as Patlak--Keller--Segel (PKS) systems were proposed in 1970s. Since
PKS systems possess relatively simple structures but admit rich dynamics,
plenty of scholars have studied them and obtained many significant results.
However, the cells or bacteria in general direct their movement in liquid. As a
consequence, it seems more realistic to consider the influence of ambient fluid
flow on the chemotactic mechanism. Motivated by this, we consider the
chemotaxis-fluid model proposed by He et al. (SIAM J. Math. Anal., Vol. 53, No.
3, 2021) in the two-dimensional bounded domain. It is well-known that the PKS
system admits the critical mass phenomenon in 2D and for the whole space
, He et al. also showed there exists the same phenomenon in the
chemotaxis-fluid system. In this paper, we first study the global
well-posedness of two-dimensional chemotaxis-fluid model in the bounded domain
and prove the solution exists globally with the subcritical mass. Then
concerning the critical mass case, we construct the boundary spot steady states
rigorously via the inner-outer gluing method. While studying the concentration
phenomenon with the critical mass, we develop the global theory of
the stationary Stokes operator in 2D
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