We prove the existence of a solution to the obstacle problem for a convection-diffusion parabolic equation whose principal part is of p-Laplacian type, with p in the subcritical range 1,2NN+2 and with lower order terms. The operator is not coercive and the obstacle function is time dependent and irregular.

We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solution to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their existence and uniqueness and show that they are also variational solutions; in particular, this is an existence result for variational solutions. Then, we apply this technique in the case of a gradient flow of a functional with inhomogeneous growth.

Stabilization and sufficient conditions for mixing by stochastic transport are shown. More precisely, given a second-order linear operator with possibly unstable eigenvalues on a smooth compact Riemannian manifold, it is shown that the inclusion of transport noise can imply global asymptotic stability. Moreover, it is shown that an arbitrary large exponential rate of convergence can be reached, implying enhanced dissipation. The sufficient conditions are shown to be satisfied by the so-called Kraichnan model for stochastic transport of passive scalars in turbulent fluids. In addition, an example is given showing that it can be sufficient to force four modes in order to induce stabilization.

We prove the existence of solutions to the Cauchy-Dirichlet problem associated with a class of doubly nonlinear anisotropic evolution equations. We also demonstrate the existence of solutions to the corresponding Cauchy problem on RN×(0,T). Under some assumptions on the Caratheodory vector field we prove a comparison principle and utilize it to obtain a uniqueness result for the Cauchy-Dirichlet problem.