Related papers: Singular and regular solutions of a non-linear par…

We consider the blow-up problem for discretized scale-invariant nonlinear dissipative wave equations. It is known that the critical exponents for undiscretized equations (continuous equations) are given by Fujita and Strauss exponents…

We study the nonlinear-damping continuation of singular solutions of the critical and supercritical NLS. Our simulations suggest that for generic initial conditions that lead to collapse in the undamped NLS, the solution of the…

We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for…

We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of…

We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic NLS. These solutions have a quartic-root blowup rate, and collapse with a quasi self-similar universal profile, which…

The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling…

We prove existence of non-commutative crepant resolutions (in the sense of van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension two, we relate these to resolutions…

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be $C^{2,1}$.

The question of whether the hyper-dissipative (HD) Napier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime--the hyper-diffusion being generated by a fractional power of the Laplacian, say…

We consider a non-Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary and with the no-slip boundary conditions for the velocity. We assume no external body forces. For the power-law like…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\eta>0$, $\beta>\frac{m\rho_1}{n-2-nm}$, $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$, $\beta_0>0$ and $\alpha_0=2\beta_0+1$. We use fixed point argument to give a new proof for the existence…

We establish new estimates for the size and structure of the nodal set $\{u=0\}$ and the singular set $\{u=|\nabla u|=0\}$ of solutions $u$ to parabolic inequalities with parabolic Lipschitz coefficients. In particular, we show that almost…

We study the existence of regular solutions of the incompressible stationary Navier-Stokes equations in $n$-dimensional Euclidean space with a given bounded external force of compact support. In dimensions $n\le 5$, the existence of such…

In this paper, for the 3-D Navier-Stokes-Boussinesq system with horizontal dissipation, where there is no smoothing effect on the vertical derivatives, we prove a uniqueness result of solutions $ (u,\rho)\in L^{\infty}_T\big( H^{0,s}\times…

This short proof shows that for smooth and sufficiently fast decaying initial data at infinity, the full incompressible Navier-Stokes solutions are eternal. The proof uses an orthogonal decomposition of the velocity field and some…

We consider the quadratic semilinear wave equation in six dimensions. This energy critical problem admits a ground state solution, which is the unique (up to scaling) positive stationary solution. We prove that any spherically symmetric…

Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the…

Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified…

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine…

We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an…