Class Of Smooth Initial Data Research Articles

Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.

On the Slightly Perturbed De Gregorio Model on $$S^1$$

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $\omega_t + a u\omega_x = u_x \omega$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a < 1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $|| \omega(t)||_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

Blow-up phenomena, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations

In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations are ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions.

NON-REGULARITY IN HÖLDER AND SOBOLEV SPACES OF SOLUTIONS TO THE SEMILINEAR HEAT AND SCHRÖDINGER EQUATIONS

In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$. We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}&gt;0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$. This yields, in particular, an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for the nonlinear Schrödinger equation in certain $H^{s}$-spaces, which depend on the smallness of $\unicode[STIX]{x1D6FC}$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel’s formula. This yields, in particular, that if $\unicode[STIX]{x1D6FC}$ is sufficiently small and $N$ is sufficiently large, then the nonlinear heat equation is ill-posed in $H^{s}(\mathbb{R}^{N})$ for all $s\geqslant 0$.

On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularity in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomena, it remains to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also gives an affirmative answer to a strong version of a problem proposed by J.Nash in 1950s.

On the Cauchy problem for the two-component Euler–Poincaré equations

In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler–Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin–Escher Lemma and Littlewood–Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.

On a one-dimensional 𝛼-patch model with nonlocal drift and fractional dissipation

We consider a one-dimensional nonlocal nonlinear equation of the form ∂ t u = ( Λ − α u ) ∂ x u − ν Λ β u \partial _t u = (\Lambda ^{-\alpha } u)\partial _x u - \nu \Lambda ^{\beta }u , where Λ = ( − ∂ x x ) 1 2 \Lambda =(-\partial _{xx})^{\frac 12} is the fractional Laplacian and ν ≥ 0 \nu \ge 0 is the viscosity coefficient. We primarily consider the regime 0 &gt; α &gt; 1 0&gt;\alpha &gt;1 and 0 ≤ β ≤ 2 0\le \beta \le 2 for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D α \alpha -patch models. In the critical and subcritical range 1 − α ≤ β ≤ 2 1-\alpha \le \beta \le 2 , we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range 0 ≤ β &gt; 1 − α 0 \le \beta &gt;1-\alpha , we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest.

On the Euler–Poincaré Equation with Non-Zero Dispersion

We consider the Euler-Poincar\'e equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae Liu}. Our analysis exhibits some new concentration mechanism and hidden monotonicity formula associated with the Euler-Poincar\'e flow. In particular we show the abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.

Exploding solutions for a nonlocal quadratic evolution problem

We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the L_x^\infty -norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczy\'nski [Biler, P. and Woyczy\'Nski, W.A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. {\bf 59} (1999), no. 3, 845-869].

On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.

Global wellposedness and blowup of solutions to a nonlocal evolution problem with singular kernels

We consider a nonlocal evolution equation in $R^2$: $\partial_t u + \nabla \cdot (u K*u )= 0$, where $K(x) = \mu \frac x {|x|^\alpha}$, $\mu=\pm 1$ and $1 < \alpha < 2 $. We study wellposedness, continuation/blowup criteria and smoothness of solutions in Sobolev spaces. In the repulsive case ($\mu=1$), by using the sharp blowup criteria, we prove global wellposedness for any positive large initial data. In the attractive case ($\mu=-1$), by using a novel free energy inequality together with a mass localization technique, we construct finite time blowups for a large class of smooth initial data.

Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation

We consider an aggregation equation in $${\mathbb {R}^n}$$ , n ≥ 2 with fractional dissipation, namely, $${u_t + \nabla\cdot(u \nabla K*u)=-\nu (-\Delta)^{\gamma/2} u}$$ , where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e −|x|. We prove that for a class of smooth initial data, the solutions develop blow-up in finite time.

Shock at very large mach number in simple gases: A physicist approach

Abstract The inviscid equations for compressible flows predict the occurence of singularities after a finite time for a large class of smooth initial data[1]. Based upon the hyperbolic nature of these equations, the method of characteristics allows a detailed study of this phenomenon in one spatial dimension. Qualitatively speaking, the formation of discontinuities is related to the fact that perturbations with a larger amplitude move faster and pile up on lower amplitude perturbations. Thus, smooth initial data may produce after a finite time infinitely sharp density and/or velocity variations. This violates a basic assumptions of continuous hydrodynamics: any field as the mass density or the fluid velocity must be almost constant on molecular scales, that is on a mean free path in gases which I shall consider from now on.