Related papers: Blowups with log canonical singularities

We fully revisit the near soliton dynamics for the mass critical (gKdV) equation. In Part I, for a class of initial data close to the soliton, we prove that only three scenario can occur: (BLOW UP) the solution blows up in finite time $T$…

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is $\frac{1}{825}$. Furthermore, the ample model $V$ achieving the minimal volume is determined uniquely up to isomorphism.…

We consider the Cauchy problem for the $L^{2}$-critical damped nonlinear Schr\"odinger equation. We prove existence and stability of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same…

We consider the focusing mass-critical nonlinear Schr\"odinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the…

We establish a new a priori estimate on solutions to the space-inhomogeneous Landau and Boltzmann equations. As a consequence, we prove a new continuation criterion, based on a weighted $L^\infty$-norm, without requiring bounds on the…

It is shown that the maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=4$, and constant dimension $k=4$ is at most $272$. In Finite Geometry terms, the maximum number of solids in…

We study mean field equations with singular sources on a compact Riemann surface with boundary $(\Sigma,g)$, subject to homogeneous Neumann boundary conditions: \[ -\Delta_g v = \rho\left( \frac{V e^{v}}{\int_\Sigma V e^{v}\, d v_g} -…

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schr\"odinger equation with inverse-square potential, \[ i\partial_t u + \Delta u + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N\geqslant 3$ and…

In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of…

While the global boundary control of nonlinear wave equations that exhibit blow-up is generally impossible, we show on a typical example, motivated by laser breakdown, that it is possible to control solutions with small data so that they…

The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can…

We consider the dyadic model with viscosity and additive Gaussian noise as a simplified version of the stochastic Navier-Stokes equations, with the purpose of studying uniqueness and emergence of singularities. We prove path-wise uniqueness…

We give a sufficient condition for blow up of positive mild solutions to an initial value problem for a nonautonomous weakly coupled system with distinct fractional diffusions. The proof is based on the study of blow up of a particular…

We generalize an intersection-theoretic local inequality of Fulton-Lazarsfeld to weighted blowups. As a consequence, we obtain the $4n^2/(k+1)$-inequality for isolated $cA_k$ singularities, an analogue of the $4 n^2$-inequality for smooth…

In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there…

The approach of the elastic continuum limit in small amorphous bodies formed by weakly polydisperse Lennard-Jones beads is investigated in a systematic finite-size study. We show that classical continuum elasticity breaks down when the…

We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is \textit{a priori} estimate…

Warped compactification of the six-dimensional bulk with a negative cosmological constant is realized with a 4-brane along with an abelian gauge theory. No fine tuning of couplings are needed to obtain the vanishing cosmological constant in…

We give an upper bound that relates the minimum weight of a nonzero componentwise product of codewords from some given number of linear codes, with the dimensions of these codes. Its shape is a direct generalization of the classical…