Related papers: Improved Blow-Up Criterion in a Variational Framew…
The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a…
We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty…
As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution…
It is shown that third-order 1D nonlinear dispersion equations admit single point gradient catastrophe, described by blow-up-type similarity solutions. After blow-up, the solutions admit shock wave-type self-similar extensions. Snce such…
The phenomenon of dissipation enhancement by transport noise is shown for stochastic 2D Navier-Stokes equations in velocity form. In the 3D case, suppression of blow-up is proved for stochastic Navier-Stokes equations in vorticity form; in…
By employing a suitable multiplicative It\^o noise with radial structure and with more than linear growth, we show the existence of a unique, global-in-time, strong solution for the stochastic Euler equations in two and three dimensions.…
In this paper we prove a blow-up criterion for the compressible Navier-Stokes-Fourier system for general thermal and caloric equations of state with inhomogeneous boundary conditions for the velocity and the temperature. Assuming only that…
This manuscript derives an evolution equation for the symmetric part of the gradient of the velocity (the strain tensor) in the incompressible Navier-Stokes equation on $\mathbb{R}^3$, and proves the existence of $L^2$ mild solutions to…
We consider the nonlinear Schr\"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same…
In this paper we will see that the global or local existence of solutions to \begin{eqnarray*} \dfrac{\partial u_{1}}{\partial t} & = & \mathit{k}_{1} (t) \Delta u_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial…
In this work we consider a system of nonlinear Schr\"odinger equations whose nonlinearities satisfy a power-type-growth. First, we prove that the Cauchy problem is local and global well-posedness in $L^2$ and $H^1$. Next, we establish the…
For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^n)}\gtrsim_{\varphi,\epsilon,(p\vee…
The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a…
Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time…
We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 <p,q< \infty$, gives rise to a strong solution with a singularity at a finite…
This work investigates radial solutions for nonlinear fractional Schr\"odinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the…
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…
This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where…
We obtain a Beale-Kato-Majda-type criterion with optimal frequency and temporal localization for the 3D Navier-Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical…
We consider the blow-up of solutions for a semilinear reaction diffusion equation with exponential reaction term. It is know that certain solutions that can be continued beyond the blow-up time possess a nonconstant selfsimilar blow-up…