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...er [[differentiability estimates|$C^{1,\alpha}$ estimates]] for variations of this situation (smooth coefficients, kernels close to the smoothness class, ...rnels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ <ref name="CS3"/>. This is enough ...

...er [[differentiability estimates|$C^{1,\alpha}$ estimates]] for variations of this situation (smooth coefficients, kernels close to the smoothness class, ...rnels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ <ref name="CS3"/>. This is enough ...

...c equations have $C^{2,\alpha}$ (therefore classical) solutions. This type of equations can be written as a Hamilton-Jacobi-Bellman equation. for a family of uniformly elliptic coefficients $a_{ij}^\alpha$. ...

A drift-(fractional)diffusion equation refers to an evolution equation of the form where $b$ is any vector field. The stationary version can also be of interest ...

The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \ ...d SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions <ref name="CMT"/>. The same comparis ...

Consider an integro-differential equation of the form ...athrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time ...

with a Fisher-KPP type of nonlinearity (for example $f(u) = u(1-u)$). In the local diffusion case, th The explanation of the difference can be understood intuitively from the fact that the fat tai ...

...s as an overview of the subject and to familiarize ourselves with the type of equations under study. ...the course is to see some regularity results for elliptic equations. Most of these results can be generalized to parabolic equations as well. However, t ...