Regular Initial Data Research Articles

Absence of dead-core formations in chemotaxis systems with degenerate diffusion

In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form ut=∇⋅(D(u)∇u−uS(u)∇v)+f(u,v),vt=Δv−uv,in a bounded domain Ω⊂RN with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient D∈C0([0,∞))∩C2((0,∞)) is assumed to satisfy D(0)=0, D(s)>0 on (0,∞), D′(s)≥0 on (0,∞) and that there are s0>0, p>1 and CD>0 such that sD′(s)≤CDD(s)andCDsp−1≤D(s)fors∈[0,s0].The sensitivity function S∈C2([0,∞)) and the source term f∈C1([0,∞)×[0,∞)) are supposed to be nonnegative.We show that for all suitably regular initial data (u0,v0) satisfying u0≥δ0>0 and v0⁄≡0 there is a time-local classical solution and – despite the degeneracy at 0 – the solution satisfies an extensibility criterion of the form eitherTmax=∞,orlim supt↗Tmax‖u(⋅,t)‖L∞(Ω)=∞.Moreover, as a by-product of our analysis, we prove that a classical solution on Ω×(0,T) obeying ‖u(⋅,t)‖L∞(Ω)≤Mu for all t∈(0,T) and emanating from initial data (u0,v0) as specified above remains strictly positive throughout Ω×(0,T), i.e. one can find δu=δu(T,δ0,Mu,‖v0‖W1,∞(Ω))>0 such that u(x,t)≥δufor all(x,t)∈Ω×(0,T).Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time.

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

Relativistic fluids in cosmological spacetimes

Abstract We review the status of mathematical research on the dynamical properties of relativistic fluids
in cosmological spacetimes - both, in the presence of gravitational backreaction as well as the
evolution on fixed cosmological backgrounds. We focus in particular on the phenomenon of fluid
stabilization, which describes the taming effect of spacetime expansion on the fluid. While fluids
are in general known to form shocks from regular initial data, spacetime expansion has been found
to suppress this behaviour. During the last decade, various rigorous results on this problem have
been put forward. We review these results, the mathematical methods involved and provide an
outlook on open questions.

Tipler naked singularities in N dimensions

Abstract A spacetime singularity, identified by the existence of incomplete causal geodesics in the spacetime, is called a (Tipler) strong curvature singularity if the volume form acting on independent Jacobi fields along causal geodesics vanishes in the approach of the singularity. It is called naked if at least one of these causal geodesics is past incomplete. Here, we study the formation of strong curvature naked singularities arising from spherically symmetric gravitational collapse of general type-I matter fields in an arbitrarily finite number of dimensions. In the spirit of Joshi and Dwivedi (1993 Phys. Rev. D 47 5357), and Goswami and Joshi (2007 Phys. Rev. D 76 084026), beginning with regular initial data, we derive two distinct (but not mutually exclusive) conditions, which we call the positive root condition (PRC) and the simple positive root condition (SPRC), that serve as necessary and sufficient conditions, respectively, for the existence of naked singularities. In doing so, we generalize the results of both the aforementioned works. We further restrict the PRC and the SPRC by imposing the curvature growth condition (CGC) of Clarke and Krolak (1985 J. Geom. Phys. 2(2) 127) on all causal curves that satisfy the causal convergence condition. The CGC then gives a sufficient condition ensuring that the naked singularities implying the PRC and implied by the SPRC, are of strong curvature type and hence correspond to the inextendibility of the spacetime. Using the CGC, we extend the results of Mosani et. al. (2020 Phys. Rev. D 101 044052) (that hold for dimension N=4) to the case N=5, showing that strong curvature naked singularities can occur in this case. However, for the case N≥6, we show that past-incomplete causal curves that identify naked singularities do not satisfy the CGC. These results shed light on the validity of the cosmic censorship conjectures in arbitrary dimensions.

Stabilization in a chemotaxis-consumption model involving Robin-type boundary conditions

The chemotaxis-consumption model{ut=∇⋅((u+1)α∇u)−∇⋅(u(u+1)β−1∇v),x∈Ω,t>00=Δv−uv,x∈Ω,t>0 is considered in a smooth bounded domain Ω⊂Rn under the boundary conditions (u+1)α∂νu=u(u+1)β−1∂νv and ∂νv=(γ−v)g on ∂Ω, with parameters α,β∈R, γ>0 and the nonnegative function g∈C1+ω(Ω¯) for some ω∈(0,1). If β−α≤1, it is proved that there exists a global bounded classical solution (u,v) for suitably regular initial data u0. Furthermore, for the given mass m=∫Ωu∞>0, the corresponding stationary system admits a unique classical solution (u∞,v∞), appearing as a large-time limit in the sense that if γ>0 is small enough, then‖u(⋅,t)−u∞‖L∞(Ω)+‖v(⋅,t)−v∞‖L∞(Ω)→0 as t→∞, with ∫Ωu∞=∫Ωu0=m>0.

Regularizing effect in a Keller–Segel system with density-dependent sensitivity for Lp-initial data of cell density

The fully parabolic Keller–Segel system{ut=Δu−χ∇⋅(u(u+1)α−1∇v)inΩ×(0,∞),vt=Δv−v+uinΩ×(0,∞) is considered in a bounded domain Ω⊂RN (N∈N) with smooth boundary, where χ,α∈R. It is shown that an associated Neumann initial-boundary value problem admits a global classical solution for initial data with low regularity, in particular, for Lp-initial data (p>max⁡{1,α,NN+1(α+1)}) of u when α<2N and when α=2N and the initial mass is small. This extends a well-known result on global solvability for reasonably regular initial data.

Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

The chemotaxis-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(n)∇c),ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇ϕ,∇⋅u=0 is considered in a smoothly bounded domain Ω⊂R2 under the boundary conditions(∇n−nχ(n)∇c)⋅ν=0,c=c⋆,u=0,x∈∂Ω,t>0 with a given nonnegative constant c⋆. It is shown that if χ∈C2([0,∞)) and χ(n)→0 as n→∞, then for all suitably regular initial data, an associated initial value problem possesses a globally defined and bounded classical solution. When ‖n0‖L1(Ω) and ‖c0‖L∞(Ω) are suitably small and c⋆≡0, we further obtain the stabilization of the classical solution.

A lower bound estimate of life span of solutions to stochastic 3D Navier–Stokes equations with convolution-type noise

In this paper, we investigate the stochastic 3D Navier–Stokes equations perturbed by linear multiplicative Gaussian noise of convolution type by transformation to random PDEs. We focus on obtaining bounds from below for the life span associated with regular initial data. The key point of the proof is the fixed point argument.

Global weak solvability in a self-consistent chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

Global weak solvability in a self-consistent chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

Effects of indirect signal absorption in the chemotaxis system involving singularly signal‐dependent motilities

AbstractWe consider the initial‐boundary problem of the chemotaxis system with indirect consumption in a smoothly bounded domain with . It is shown that for all suitably regular initial data, global solvability in classical sense can be established even with singularly signal‐dependent motilities involved. To be specific, the global classical solution is constructed for arbitrary .

BMS-supertranslation charges at the critical sets of null infinity

For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity I− can be related to those at future null infinity I+ via an antipodal map at spatial infinity i0. We analyze the validity of this conjecture using Friedrich’s formulation of spatial infinity, which gives rise to a regular initial value problem for the conformal field equations at spatial infinity. A central structure in this analysis is the cylinder at spatial infinity I representing a blow-up of the standard spatial infinity point i0 to a 2-sphere. The cylinder I touches past and future null infinities I± at the critical sets I±. We show that for a generic class of asymptotically Euclidean and regular initial data, BMS-supertranslation charges are not well-defined at I± unless the initial data satisfies an extra regularity condition. We also show that given initial data that satisfy the regularity condition, BMS-supertranslation charges at I± are fully determined by the initial data and that the relation between the charges at I− and those at I+ directly follows from our regularity condition.

Global solvability for an indirect consumption chemotaxis system with signal-dependent motility

This paper considers the indirect signal consumption-chemotaxis system with signal-dependent motility in a smooth bounded domain Ω⊂Rn(n≤3), as given by ut=Δ(ϕ(v)u),vt=Δv−vw,wt=dΔw−w+u, where the motility function ϕ∈C3((0,∞)),ϕ>0 on (0,∞), which generalizes ϕ(v)=vα,α∈R. Based on point-wise positive lower bound estimate of v, it is shown that for any suitably regular initial data, the corresponding initial–boundary value problem admits global smooth solutions.

Global existence and boundedness in a two-species chemotaxis-fluid system with indirect pursuit–evasion interaction

Abstract This paper investigates a two-species chemotaxis-fluid system with indirect pursuit–evasion interaction in a bounded domain with smooth boundary. Under suitably regular initial data and no-flux/no-flux/no-flux/no-flux/Dirichlet boundary conditions, we prove that the system possesses a global bounded classical solution in the two-dimensional and three-dimensional cases. Our results extend the result obtained in previously known ones and partly result is new.

Global solvability and stabilization in a three-dimensional coral fertilization model involving the Navier-Stokes equations

This paper deals with a coral fertilization model involving the Navier-Stokes equations. It is proved that an associated initial-boundary value problem with no-flux, Dirichlet boundary conditions in a three-dimensional smoothly bounded domain admits a globally defined weak solution for any appropriately regular initial data whenevermax⁡{‖c0‖∞,‖m0‖∞}<π2χ, where χ is the chemotaxis sensitivity and m0,c0 are initial density of unfertilized eggs and initial concentration of the chemical expelled by the eggs, respectively. Moreover, it is showed that the obtained solutions stabilize to a certain constant equilibrium.

Global Regularity for Gravity Unstable Muskat Bubbles

In this paper, we study the dynamics of fluids in porous media governed by Darcy’s law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.

How far do indirect signal production mechanisms influence regularity in the three-dimensional Keller–Segel–Navier–Stokes system?

This paper reconsiders the Keller–Segel–Navier–Stokes model with indirect signal production and weak logistic-type degradation in a three-dimensional (3D) bounded and smooth domain. One recent literature has asserted that for all reasonably regular initial data, the associated initial-boundary value problem with certain sub-quadratic degradation possesses at least one global generalized solution which is uniformly bounded and converges to the constant equilibrium in [Formula: see text] with any [Formula: see text]. Nevertheless, the knowledge on regularity properties of solution has not yet exceeded some information on fairly basic integrability features. The present study firstly elevates the regularities of some solution components under the same assumption on degradation exponent, especially establishes the global boundedness in [Formula: see text] with [Formula: see text], and secondly asserts that each of these generalized solutions becomes eventually classical and bounded in [Formula: see text] under some suitably strong sub-quadratic degradation assumption and an explicit smallness condition on the intrinsic growth rate of cells relative to some powers of the degradation coefficient. As a by-product of the latter, these solutions are shown to stabilize toward the corresponding spatially homogeneous state in [Formula: see text]. Our results indeed provide a more in-depth understanding on the global dynamics of solutions, and significantly improve previously known ones. In comparison to the related contributions in the case of direct signal production, our findings inter alia rigorously reveal that the indirect signal production mechanism results in some genuine regularizing effects in the sense that the possibly destabilizing action of chemotactic cross-diffusion in the 3D Keller–Segel–Navier–Stokes system can be greatly weakened or even completely excluded by this indirectness.

Nonlinear transmission exponent for boundedness of solutions to a chemotaxis system with indirect signal production

This paper deals with the Neumann initial–boundary problem for a fully parabolic chemotaxis system with indirect signal production, and the size of phenotypic transmission exponent is identified to guarantee the global existence and boundedness of classical solutions to the problem with suitably regular initial data.

Global classical solvability and asymptotic behaviors of a parabolic-elliptic Chemotaxis-type system modeling crime activities

This paper is devoted to a two-dimensional parabolic-elliptic system in crime activities. The main difference between this system and the chemotaxis system with logarithmic singular sensitivity and quadratic logistic source is that the source term of this system is a mixtype quadratic term. The local existence of solutions to its intial boundary valve problem associated with homogeneous Neumann boundary conditions was considered by Rodríguez. The present study indicates that for any properly regular initial data there admits global classical solutions. Moreover, the qualitative behaviors of solutions in large time scales are considered as well.

Eventual smoothness and asymptotic stabilization in a two-dimensional logarithmic chemotaxis-Navier–Stokes system with nutrient-supported proliferation and signal consumption

In this paper, we study the influence of nutrient-dependent cell proliferation on large time behavior of the solutions to an associated initial-boundary problem of(⋆){nt+u⋅∇n=Δn−∇⋅(nc∇c)+nc,x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+(u⋅∇)u=Δu+∇P+n∇Φ,x∈Ω,t>0, subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain Ω⊂R2, where +nc in the first equation denotes cell reproduction in dependence on nutrients. It is proved that the initial-boundary problem is globally solvable in a generalized sense for any appropriately regular initial data, and that the generalized solutions thereof emanating from suitably small initial data will become smooth from a finite waiting time and exponentially approach (1|Ω|∫Ω(n0+c0),0,0) as t→∞.As compared to a precedent result obtained without cell proliferation, the complexity caused by +nc in the first equation requires some conditions on the initial data c0 rather only on n0 in order to derive the large time behavior of the solutions thereof, which seems comprehensible because of the necessity for offsetting the possibly increasing trend of cell by +nc.

The Existence of Entropy Solutions for a Class of Parabolic Equations

The existence and uniqueness of entropy solutions for a class of parabolic equations involving a p(x)-Laplace operator are investigated. We first prove existence of the global weak solution for the p(x)-Laplacian equations with regular initial data via the difference and variation methods as well as the standard domain expansion technique. Then, by constructing and solving a related approximation problem, the entropy solution for the p(x)-Laplacian equations with irregular initial data in whole space is also obtained.