  The K7 T Tauri star PDS 70 remains the best laboratory for investigating the
influence of giant planet formation on the structure of the parental disk. One
of the most intriguing discoveries is the detection of a resolved inner disk
from ALMA observations that extends up to the orbit of PDS 70b. It is
challenging to explain this inner disk because most of the dust particles are
expected to be trapped at the outer edge of the gap opened by PDS 70b and PDS
70c. By performing dust evolution models in combination with radiative transfer
simulations that match the gas disk masses obtained from recent thermo-chemical
models of PDS 70, we find that when the minimum grain size in the models is
larger than 0.1$\mu$m, there is an efficient filtration of dust particles, and
the inner disk is depleted during the first million year of dust evolution. To
maintain an inner disk, the minimum grain size in the models therefore needs to
be smaller than 0.1$\mu$m. Only when grains are that small are they diffused
and dragged along with the gas throughout the gap opened by the planets. The
small grains transported in the inner disk grow and drift into it, but the
constant reservoir of dust particles that are trapped at the outer edge of the
gap and that continuously fragment allows the inner disk to refill on
million-year timescales. Our flux predictions at millimeter wavelength of these
models agree with ALMA observations. These models predict a spectral index of
3.2 in the outer and 3.6 in the inner disk. Our simple analytical calculations
show that the water emission in the inner disk that was recently observed with
the James Webb Space Telescope may originate from these ice-coated small grains
that flow through the gap, grow, and drift toward the innermost disk regions to
reach the water snowline.
