Related papers: A note on local higher regularity in the dynamic l…
We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of $\Delta u -u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue point of…
This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…
Fluid discontinuities, such as shock fronts and vortex sheets, can reflect waves and become unstable to corrugation. Analytical calculations of these phenomena are tractable in the simplest cases only, while their numerical simulations are…
We prove that the minimizer in the N\'ed\'elec polynomial space of some degree p > 0 of a discrete minimization problem performs as well as the continuous minimizer in H(curl), up to a constant that is independent of the polynomial degree…
This paper proposes a relaxed control regularization with general exploration rewards to design robust feedback controls for multi-dimensional continuous-time stochastic exit time problems. We establish that the regularized control problem…
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…
The aim of this article is to investigate the well-posedness, stability and convergence of solutions to the time-dependent Maxwell's equations for electric field in conductive media in continuous and discrete settings. The situation we…
We prove a logarithmic improvement of the Caffarelli-Kohn-Nirenberg partial regularity theorem for the Navier-Stokes equations. The key idea is to find a quantitative counterpart for the absolute continuity of the dissipation energy using…
We prove a regularity result for the unstable elliptic free boundary problem $\Delta u = -\chi_{\{u>0\}}$ related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are…
Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Yi et. al. (2021) has demonstrated the existence of such waves for two classes of regularisations, including…
We study equations from the area of peridynamics, which is an extension of elasticity. The governing equations form a system of nonlocal wave equations. Its governing operator is found to be a bounded, linear and self-adjoint operator on a…
We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily…
We lay down a geometric-analytic framework to capture properties of energy dissipation within weak solutions to the incompressible Euler equations. For solutions with spatial Besov regularity, it is proved that the Duchon-Robert…
Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary…
We study the shape stability of disks moving in an external Laplacian field in two dimensions. The problem is motivated by the motion of ionization fronts in streamer-type electric breakdown. It is mathematically equivalent to the motion of…
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic…
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time,…
A wave front and a wave back that spontaneously connect two hyperbolic equilibria, known as a heteroclinic wave loop, give rise to periodic waves with arbitrarily large spatial periods through the heteroclinic bifurcation. The nonlinear…
We study local regularity for nonlocal doubly degenerate parabolic equations. The model equation is \begin{equation*}\begin{split}…
We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$, where $\Omega\subset…