Mass Concentration Phenomenon Research Articles

Mass concentration phenomenon in the 3D bipolar compressible Navier–Stokes–Poisson system

Mass concentration phenomenon in the 3D bipolar compressible Navier–Stokes–Poisson system

Sharp Threshold of Global Existence and Mass Concentration for the Schrödinger–Hartree Equation with Anisotropic Harmonic Confinement

This article is concerned with the initial-value problem of a Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement. First, we get a sharp threshold for global existence and finite time blow-up on the ground state mass in the L 2 -critical case. Then, some new cross-invariant manifolds and variational problems are constructed to study blow-up versus global well-posedness criterion in the L 2 -critical and L 2 -supercritical cases. Finally, we research the mass concentration phenomenon of blow-up solutions and the dynamics of the L 2 -minimal blow-up solutions in the L 2 -critical case. The main ingredients of the proofs are the variational characterisation of the ground state, a suitably refined compactness lemma, and scaling techniques. Our conclusions extend and compensate for some previous results.

Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2

Abstract We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ 2 {\mathbb{R}^{2}} , where the intra-component interactions μ i {\mu_{i}} and the inter-component interactions β i ⁢ j = β j ⁢ i {\beta_{ij}=\beta_{ji}} ( i , j = 1 , 2 , 3 {i,j=1,2,3} , i ≠ j {i\neq j} ) are all attractive. We display the regions of μ i {\mu_{i}} and β i ⁢ j {\beta_{ij}} for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as β i ⁢ j ↗ β i ⁢ j * := a ∗ + 1 2 ⁢ ( a ∗ - μ i ) ⁢ ( a ∗ - μ j ) {\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i% })(a^{\ast}-\mu_{j})}}} , where 0 < μ i < a ∗ := ∥ w ∥ 2 2 {0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}} are fixed and w is the unique positive solution of Δ ⁢ w - w + w 3 = 0 {\Delta w-w+w^{3}=0} in H 1 ⁢ ( ℝ 2 ) {H^{1}(\mathbb{R}^{2})} . The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.

Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation

We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case \(K(x)=-|x|\). We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity \(\varepsilon \) studied in our previous paper (Biler et al. in J Differ Equ 271:1092–1108, 2021), obtaining sharp upper and lower bounds for Sobolev norms.

Concentration phenomena in a diffusive aggregation model

We consider the drift-diffusion equation ut−εΔu+∇⋅(u∇K⁎u)=0 in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied.

Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations

This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far field density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in finite time, and it is independent of the velocity and magnetic field. In particular, this extends the corresponding Du's et al. results (Nonlinearity, 28, 2959-2976, 2015, [4]) to bounded domain in $ \mathbb{R}^2 $ when the initial density and the initial magnetic field are decay not too show at infinity, and Ji's et al. results (Discrete Contin. Dyn. Syst., 39, 1117-1133, 2019, [10]) to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic field.

Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition

We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has negative free energy, then either the corresponding weak solution to the equation does not exist globally in time, or the time global solution does not remain bounded in the energy space. We emphasize that our result does not require any weight assumption on the initial data, and hence, a solution may have an infinite second moment. The proof is based upon the modified virial law and conservation laws and we show that the modified moment functional vanishes for a finite time under the negative energy condition. For a radially symmetric case, the solution blows up in finite time and the mass concentration phenomenon occurs with a sharp lower bound related to the best constant for the Hardy–Littlewood–Sobolev inequality.

Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations

In this paper, we consider the global strong solutions to the Cauchy problem of the compressible Navier-Stokes equations in two spatial dimensions with vacuum as far field density. It is proved that the strong solutions exist globally if the density is bounded above. Furthermore, we show that if the solutions of the two-dimensional (2D) viscous compressible flows blow up, then the mass of the compressible fluid will concentrate on some points in finite time.

The rate of concentration for the radially symmetric solution to a degenerate drift-diffusion equation with the mass critical exponent

We consider the concentration rate of the total mass for radially symmetric blow-up solutions to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. We proved that the radially symmetric solution blows up in finite time when the initial data has negative free energy. We show that the mass concentration phenomenon occurs with the sharp lower constant related to the best constant of the Hardy–Littlewood–Sobolev inequality and the concentration rate of the total mass.

The L2 Weak Sequential Convergence of Radial Focusing Mass Critical NLS Solutions with Mass Above the Ground State

Abstract We study the non-scattering $L^{2}$ solution $u$ to the radial focusing mass-critical nonlinear Schrödinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration phenomena of the minimal mass blow-up solution.

Global wellposedness for a one-dimensional Chern–Simons–Dirac system in Lp

ABSTRACTThe global wellposedness in Lp(ℝ) for the Chern–Simons–Dirac equation in the 1+1 space and time dimension is discussed. We consider two types of quadratic nonlinearity: the null case and the non-null case. We show the time global wellposedness for the Chern–Simon–Dirac equation in the framework of Lp(ℝ), where 1≤p≤∞ for the null case. For the scaling critical case, p = 1, mass concentration phenomena of the solutions may occur in considering the time global solvability. We invoke the Delgado–Candy estimate which plays a crucial role in preventing concentration phenomena of the global solution. Our method is related to the original work of Candy (2011), who showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in the critical space L2(ℝ).

The mass concentration phenomenon for L 2-critical constrained problems related to Kirchhoff equations

In this paper, we study the concentration behavior of critical points with a minimax characterization to the following functional $$I(u)=\frac{a}{2} \int\limits_{{\mathbb{R}}^N}|\nabla u|^2+\frac{b}{4} \left(\ \int\limits_{{\mathbb{R}}^N}|\nabla u|^2\right)^2-\frac{N}{2N+8} \int\limits_{{\mathbb{R}}^N}|u|^{\frac{2N+8}{N}}$$ constrain on \({S_c=\{u\in H^1({\mathbb{R}}^N)|~|u|_2=c,c > 0\}}\) when \({c\rightarrow (c^*)^+}\), where \({c^*=\left(2^{-1}b|Q|_2^{\frac{8}{N}} \right)^{\frac{N}{8-2N}}}\), \({N=1,2,3,}\) and \({Q}\) is up to translations, the unique positive solution of \({-2\Delta Q+\left(\frac{4}{N}-1\right)Q=|Q|^{\frac{8}{N}} Q}\) in \({{\mathbb{R}}^N}\).

Mass concentration phenomenon in compressible magnetohydrodynamic flows

In this paper, we investigate the singularity formulation and blow-up mechanism to the compressible magnetohydrodynamic (MHD) flows in two dimensions. It is essentially shown that the mass of the compressible MHD fluid will concentrate in some spatial points, even if the initial density contains vacuum states, provided that the strong or smooth solution develops singularity in finite time. The result indicates that there is no possibility of formulating the other kinds of singularities, such as the vanishing of vacuum states, the appearance of vacuum states in the non-vacuum region or even milder ones, before the density concentrates at some point. It also implies the global regularity of the strong solution to the compressible MHD system, provided that the density remains bounded.

Blow-up with mass concentration for the long-wave unstable thin-film equation

For a family of long-wave unstable thin-film equations, we prove existence of non-negative weak solutions blowing-up in a finite time. Specifically, building these solutions from initial data with negative energy, we show that their -norms go to infinity as . In addition, using the Bourgain’s type approach, we obtain qualitative information about the blow-up and prove mass concentration phenomenon.

Mass concentration phenomenon for inhomogeneous fractional Hartree equations

In this paper, we consider the inhomogeneous fractional Hartree equation in mass-critical case. First, we prove the existence of the ground states by studying the related constrained minimization problem. When the mass of initial data is larger than the mass of the ground states, we construct a sequence of blow-up solutions whose initial data converge to the ground states. Moreover, we show the mass concentration phenomenon of the blow-up solutions via applying the concentration-compactness principle.

Anisotropic diffusion in oriented environments can lead to singularity formation

We consider an anisotropic diffusion equation of the form ut = ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

MASS CONCENTRATION WITH MIXED NORM FOR A NONELLIPTIC SCHRÖDINGER EQUATION

AbstractThis paper is concerned with a mass concentration phenomenon for a two-dimensional nonelliptic Schrödinger equation. It is well known that this phenomenon occurs when the ${L}^{4} $-norm of the solution blows up in finite time. We extend this result to the case where a mixed norm of the solution blows up in finite time.

Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schrodinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in L^2$ for $\alpha >2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $\alpha$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.

On Mass Concentration for the L 2-Critical Nonlinear Schrödinger Equations

We consider the mass concentration phenomenon for the L 2-critical nonlinear Schrödinger equations. We show the mass concentration of blow-up solutions contained in space near the finite time. The new ingredient in this paper is a refinement of Strichartz's estimates with the mixed norm for 2 < q ≤ r.

The Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System

We study the gravitational Vlasov Poisson system \({f_t + v \cdot \nabla_{x} f - E \cdot \nabla_{v} f = 0}\) = 0 where \({E(x) = \nabla_{x}\phi(x)}\), \({\Delta_{x}\phi = \rho(x)}\), \({\rho(x) = \int_{\mathbb{R}^{N}} f(x, v)\,{\rm d}x\,{\rm d}v}\), in dimension N = 3, 4. In dimension N = 3 where the problem is subcritical, we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space. This implies, in particular, the orbital stability in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in [11, 16, 19]. In dimension N = 4 where the problem is L1 critical, we obtain the polytropic steady states as best constant minimizers of a suitable Sobolev type inequality relating the kinetic and the potential energy. We then derive using an explicit pseudo-conformal symmetry the existence of critical mass finite time blow-up solutions, and prove more generally a mass concentration phenomenon for finite time blow up solutions. This is the first result of description of a singularity formation in a Vlasov setting. The global structure of the problem is reminiscent of the one for the focusing nonlinear Schrödinger equation iu t = −Δu−|u|p−1u in the energy space \({H^1(\mathbb{R}^N)}\) .