Whether the multi-dimensional Euler-Poisson system admits global smooth solutions remains a challenging open problem. In this paper, we construct a class of large-data global smooth solutions to the one-fluid Euler-Poisson system in ℝ^d (1≤ d≤ 5) by using the relaxation dissipation mechanism. Precisely, assuming that the initial density is far from vacuum and ε E_0 is sufficiently small, where E_0 denotes the initial energy and ε is the relaxation time, we establish the global well-posedness of smooth solutions to the Cauchy problem. In particular, the size of the initial perturbation may be arbitrarily large, provided that the relaxation time is sufficiently small. Furthermore, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate 𝒪(e^-λtε) between the rescaled Euler-Poisson system and the limiting drift-diffusion system for ill-prepared data. The new ingredient lies in developing the maximum principle for the nonlinear drift-diffusion system with nonlocal effect, which leads to the large-data global existence.

We are concerned with the Cauchy problem of the compressible quantum Euler system with damping in ℝ^d (d≥ 1) . Compared with [12], we establish the local well-posedness of solutions near constant equilibrium in the critical regularity functional framework. Furthermore, we develop a new craftsmanship condition for generally hyperbolic systems with symmetric damping and Korteweg-type dispersion, which enables us to prove the global-in-time well-posedness within the regime of small data. The so-called “gauge” function technique is mainly employed to overcome the possible derivative loss in energy estimates. Finally, we derive optimal time-decay estimates of solutions by the time-weighted Lyapunov energy method, under the assumption that the low-frequency part of the initial perturbation is bounded in Ḃ^-σ _1_2,∞ with σ _1∈ (-d/2, d/2] .

We investigate the Cauchy problem for the Navier–Stokes equations for viscous compressible fluids with capillarity. The linear third-order capillarity term behaves like the heat diffusion of density fluctuations, which enables us to provide an equivalent characterization of Gevrey analyticity, optimal decay, and initial regularity criteria in ℝ^d ( d ≥ 2 ). Precisely, in the critical L^p framework, it is proved that the Besov space Ḃ^σ _1_2,∞ -boundedness condition (with d/2-2d/p≤σ _1

In this paper, we consider the three-dimensional incompressible rotating Navier–Stokes equations and establish the sharp L^p decay estimates of global solutions. We reveal that the optimal L^p decay rates for 2<p<∞ are strictly faster than those obtained in existing results by interpolation between the L^2 unitary identity and L^∞ dispersive estimates, although the endpoint cases were known to be sharp. Moreover, the optimality of decay rates is also proved by the lower bound estimate for a specific initial datum. The underlying mechanism lies in the anisotropic degeneracy of the oscillatory integrals arising from the Coriolis force.

We present a new derivation for the optimal decay of arbitrary higher order derivatives for Lp solutions to the compressible fluid model of Korteweg type. This approach, based on Gevrey estimates, is to establish uniform bounds on the growth of the radius of analyticity of the solution in negative Besov norms. For that end, the maximal regularity property involving Gevrey multiplier of heat kernel and non standard product Besov estimates are well developed. Our approach is partly inspired by Oliver-Titi’s work and is applicable to a wide range of dissipative systems.

We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in ℝ^d ( d≥ 1 ), where the interaction force is given by ∇ (-Δ )^(α -d)/2ρ with d-2<α

We investigate the three-dimensional compressible Euler-Maxwell system, a model for simulating the transport of electrons interacting with propagating electromagnetic waves in semiconductor devices. First, we establish the global well-posedness of classical solutions near constant equilibrium in a critical regularity setting, uniformly with respect to the relaxation parameter epsilon>0. Then, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate O(epsilon) between the rescaled Euler-Maxwell system and the limiting drift-diffusion model. This provides the first global-in-time strong convergence result for the relaxation procedure in the case of ill-prepared data so far. We propose a new characterization of the dissipation structure for the non-symmetric relaxation of linearized Euler-Maxwell system, which partitions the frequency space into three distinct regimes (low, medium and high frequencies) associated with different behaviors of the solution. Within each regime, the application of Lyapunov functionals based on the hypocoercivity theory reveals the expected dissipative properties. Moreover, two correction functions are employed to take care of the initial layers in the relaxation convergence.

In this paper, we mainly study the blow up phenomenon to classical solutions of compressible non-isentropic Euler equations with time-dependent damping a/(1+t)(lambda) u in one space dimension. By constructing the decoupled Riccati equation, we show that C1 solutions will blow up in finite time when the adiabatic gas constant 1 < gamma < 3 and the damping coefficient lambda >= 0 if the initial data satisfies suitable condition. Moreover, when the initial data is small enough, we can see that the blow up comes from derivatives of the solution. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study the unique solvability of weak transmission problems in some unbounded domains containing at least one flat layer area, which is associated with the motion of two-phase fluids. In particular, we construct the solution to the transmission problem for the Laplace operator with non-homogeneous boundary conditions. As a direct consequence, the Helmholtz–Weyl decomposition for the two-phase problem is also proved.

This work is dedicated to the vanishing capillarity limit of strong solutions to the compressible Navier-Stokes-Korteweg system in Lp Besov spaces. We constructed the uniform global-in-time solutions with respect to the capillarity parameter kappa by employing the viscous effective flux. The strong solution satisfies the asymptotic formula (& rhov;kappa, u kappa) = (& rhov;0, u0) + O(root kappa) in the critical regularity setting for any positive times, where (& rhov;0, u0) is the global solution to the compressible Navier-Stokes system. This is the first effort to show the global vanishing capillarity limit at the convergence rate of order root kappa.

We construct a unique global solution with critical regularity to the Cauchy problem for the three-dimensional Boltzmann equation with initial data near the Maxwellian in the spatially critical Besov space L-2 xi (B-2,1(1/2))boolean AND L-2 xi (B-2,1(3/2)) Furthermore, under the condition that the low-frequency part of the initial perturbation is bounded in L-2 xi (B-sigma 0 (2),(infinity) ) with - 3/2 sigma 0 0<1/2, it is shown that the solution converges to equilibrium in large times with the optimal rate of O (t - (sigma -sigma(0))/2) in L-2 xi (B-2,1(3/2))for sigma -sigma(0), and the microscopic part decays at an enhanced rate of O (t - ((sigma -sigma(0))/2 - 1/2) This is the first work to address the global existence and large-time behavior of solutions to the Boltzmann equation in the homogeneous critical setting We develop the hypocoercivity theory, which indicates Lyapunov functionals with different dissipation rates in low and high frequencies Moreover, the so-called time-weighted Lyapunov energy argument is employed to obtain the optimal time-decay estimates

In this paper, we justify the validity of global-in-time convergence rigorously from the pressureless relaxed Euler-Riesz system to the frictional porous medium model in Rd (d ≥ 1). For initial data close to a constant equilibrium in critical Besov spaces, we prove the global existence and uniqueness of strong solutions for the pressureless damped Euler-Riesz system, uniformly with respect to the relaxation parameter. Furthermore, we derive global-in-time error estimates between solutions of the pressureless Euler-Riesz system and the porous medium model for ill-prepared data, with an explicit convergence rate. The proof relies on the non-local hypocoercivity argument, which enables us to track the regularity evolution with respect to the relaxation parameter and exhibit the frictional dissipative mechanism arising from Riesz interactions.

The partially dissipative systems that characterize many physical phenomena were first pointed out by Godunov (1961), then investigated by Friedrichs-Lax (1971) who introduced the convex entropy, and later by Shizuta-Kawashima (1984,1985) who initiated a simple sufficient criterion ensuring the global existence of smooth solutions and their large-time asymptotics. There has been remarkable progress in the past several decades, through various different attempts. However, the decay character theory for partially dissipative hyperbolic systems remains largely open, as the Fourier transform of Green's function is generally not explicit in multi-dimensions. In this paper, we provide a positive answer to the open question by means of the general L^p energy method. Precisely, a new effective quantity Ψ(t,x) motivated by the compressible Euler system with damping is introduced, which enables us to capture leading diffusion profiles of the large-time behavior in the spirit of the Chapman-Enskog expansion. Consequently, we prove that the solutions approach the constant equilibrium state in the Ḃ^σ_p,1-norm at the rate t^-(σ-σ_1)/2 as t→∞, and the corresponding norm of dissipative components decays at the enhanced rate t^-(σ-σ_1+1)/2, where the boundedness assumption in the Ḃ^σ_1_p,∞ (-d/p≤ σ_1<d/p-1)-norm of the low frequencies of conservative components is not only sufficient, but also necessary to achieve those upper bounds of decay estimates. Furthermore, both upper and lower bounds for time-decay estimates are obtained if and only if the low-frequency part of Ψ_0(x) (the initial effective quantity) is bounded in a non-trivial subset of Ḃ^σ_1_p,∞.

We investigate the Navier–Stokes–Cattaneo–Christov (NSC) system in ℝ^d ( d≥ 3 ), a model of heat-conductive compressible flows serving as a finite speed of propagation approximation of the Navier–Stokes–Fourier (NSF) system. Due to the presence of Oldroyd’s upper-convected derivatives, the system (NSC) exhibits a lack of hyperbolicity which makes it challenging to establish its well-posedness, especially in multi-dimensional contexts. In this paper, within a critical regularity functional framework, we prove the global-in-time well-posedness of (NSC) for initial data that are small perturbations of constant equilibria, uniformly with respect to the approximation parameter ε >0 . Then, building upon this result, we obtain the sharp large-time asymptotic behaviour of (NSC) and, for all time t>0 , we derive quantitative error estimates between the solutions of (NSC) and (NSF). To the best of our knowledge, our work provides the first strong convergence result for this relaxation procedure in the three-dimensional setting and for ill-prepared data. The (NSC) system is partially dissipative and incorporates both partial diffusion and partial damping mechanisms. To address these aspects and ensure the large-time stability of the solutions, we construct localized-in-frequency perturbed energy functionals based on the hypocoercivity theory. More precisely, our analysis relies on partitioning the frequency space into three distinct regimes: low, medium and high frequencies. Within each frequency regime, we introduce effective unknowns and Lyapunov functionals, revealing the spectrally expected dissipative structures.

The low-frequency L 1 assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible NavierStokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical L p framework. Precisely, it is proved that the Besov space boundedness condition (with d 2 - 2dp dp = s 1 < d 2 - 1) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of B ? s 1 2,8. , 8 . To the best of our knowledge, our B ? s 1 2,8- , 8- work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Cooperative navigation realized through information interaction among UAVs can improve the navigation performance of UAVs in complex environments. In this paper, we construct a distributed cooperative navigation framework that fuses measurements from various onboard navigation sensors and inter-UAV ranging. Considering the computational efficiency, we propose a Gaussian particle filter for message passing and computation, which reduces the computational load. Flight test results show that the algorithm can significantly improve the computational efficiency with the same positioning accuracy compared to the traditional BP algorithm.

We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in L p L^p -type critical Besov spaces. In comparison with the results obtained by Crin-Barat and Danchin [Pure Appl. Anal. 4 (2022), pp. 85–125; Math. Ann. 386 (2023), pp. 2159–2206], the more general pressure law satisfying P ′ ( ρ ¯ ) > 0 P’(\bar {\rho })>0 is allowed. To achieve it, a new composition estimate is established in the L 2 L^2 - L p L^p hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies.

We are concerned with a system of equations governing the evolution of isothermal, viscous, and compressible fluids of the Korteweg type, which is used to describe a two-phase liquid–vapor mixture. It is found that there is a “regularity-gain" dissipative structure of linearized systems in case of zero sound speed P′(ρ*)=0, in comparison with the classical compressible Navier–Stokes equations. First, we establish the global-in-time existence of strong solutions in hybrid Besov spaces by using Banach’s fixed point theorem. Furthermore, we prove that the global solutions with critical regularity are Gevrey analytic in fact. Secondly, based on Gevrey’s estimates, we obtain uniform bounds on the growth of the analyticity radius of solutions in negative Besov spaces, which lead to the optimal time-decay estimates of solutions and their derivatives of arbitrary order.

We are concerned with a system of equations in Rd (d > 3) governing the evolution of isothermal, viscous and compressible fluids of Korteweg type, that can be used as a phase transition model. In the case of zero sound speed P'(& rho;*) = 0, it is found that the linearized system admits the purely parabolic structure, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of Lp-type. Precisely, if the full viscosity coefficient and capillary coefficient satisfy & nu; over bar 2 > 4 & kappa; over bar , then the acoustic waves are not available in compressible fluids. Consequently, the prior L2 boundedness on the low frequencies of density and velocity could be improved to the general Lp version with 1 < p < d. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions.& COPY; 2023 Elsevier Inc. All rights reserved.

This paper is dedicated to the study of viscous compressible liquid-gas two-phase flow model in multi-dimensional spaces d >= 2. We investigate the global existence of strong solutions to the Cauchy problem in the L-p critical regularity framework. In comparison with previous results, the high-frequency of dissipative variable ((P) over tilde, (u) over tilde) may be bounded in the critical space (B)over dot(p,1)(d/p) x (B)over dot(p,1)(d/p-1) (p >= 2) and the high-frequency of non-dissipative variable (rho) over tilde to be bounded in the L-2-type Besov space (B)over dot(2,1)(d/2). Moreover, a Lyapunov-type energy argument can be developed, which leads to the time-decay estimates of solutions without additional smallness assumptions.