ncts

- \partial_{j} (a_{i j} (x) \partial_{i} u) + b_{i} (x) \partial_{i} u + c (x) u = f (x)

where the drift coefficients

a_{i j}

are bounded and elliptic, but may not be continuous. In this minicourse we shift our focus to critical drift coefficients

b \in L^{n}

b \in L_{w e a k}^{n}

with div

b \geq 0

. We will consider the existence and

L^{q}

gradient estimates for

q

-weak solutions with critical drift in a bounded domain in

R^{n}

n \geq 2

, and the same problems for stationary perturbed Stokes systems with divergence-free critical drift

- Δ u + b (x) \cdot \nabla u + \nabla p = f (x),

div

u =

div

b = 0

These are motivated by open questions for stationary Navier-Stokes equations.

For dimension

n = 2

or higher, these results are proved for any finite exponent

q > 1

assuming the drift is either in

L^{n}

or sufficiently small in weak

L^{n}

, by perturbative arguments.

The more significant case is large drift in weak

L^{n}

. For scalar PDE and

n \geq 3

, these results are proved for

n^{'} < q < n

, and Hölder continuity is shown for

q > n

. For scalar PDE and

n = 2

, or for perturbed Stokes systems for

n \geq 2

, these results are proved for

q

sufficiently close to

2

, depending on the drift size. The dimension

2

case requires special care as the energy estimate for

2

-weak solutions is not defined.

2. Outline & Descriptions

Thefollowingisthetentativelistoftopics:

(1) Introduction:statements,conjectures,andcriticality

(2) De Giorgiand Moser’s methods

(3) Lorentzspaces:interpolation,H¨olderinequality,andimbedding

(4) EstimatesforscalarPDEwithcriticaldrift,n≥3

(5) Gehring’smethodofreverseH¨olderinequality

(6) Wolf’slocalpressuredecomposition

(7) EstimatesforperturbedStokessystems

3. Reference
[Gia93] Mariano Giaquinta. Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, 1993.[HL11] Qing Han and Fanghua Lin. Elliptic partial dierential equations, volume 1 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, second edition, 2011.

4. Registration

https://forms.gle/CX5kZwqgdhWdSjPM7

5. Online Meeting Link

https://us02web.zoom.us/j/82182146536?pwd=rq9I4LHVbcH569bfNg7LWlARj5LVsQ.1