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This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a Lp–Ls type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrt{\rho} \in L^\infty_t H^1_x$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.

In emergency decision making, it can be difficult for decision-makers (DMs) to identify all possible scenarios due to a lack of information and the evolution of emergency situations. Therefore, this paper presents an incomplete probabilistic linguistic term set (InPLTS), which is a generalized hesitant fuzzy linguistic term set (HFLTS). The InPLTS can more appropriately describe a case in which a DM considers several possible linguistic terms with uncertain probabilities. Furthermore, this work extends the InPLTS to an incomplete probabilistic linguistic preference relation (InPLPR) and proposes a complete algorithm based on an emergency fault tree analysis (EFTA) to estimate missing entries of the InPLPR. The work also investigates the expected consistency, acceptable expected consistency, and consistency-improving methods for the reasonable application of the InPLPR. Then, a consistency-based emergency decision-making method using the InPLPR is proposed to address issues related to a lack of information, uncertainties and dynamic trends. In using this method, DMs can evaluate emergency alternatives of different possible scenarios with the InPLPR, and the impacts of different emergency responses on the evolution of emergencies can also be considered. Finally, the InPLPRs and the abovementioned method are applied to a public health emergency decision-making process to illustrate the advantages of the proposed method.

This paper deals with the compressible, viscous and heat-conducting magnetohydrodynamic (MHD) equations in dimension one. In the case when all transport coefficients are positive constant, we establish the global existence and uniqueness of strong solution for the initial–boundary-value problem. The initial data is large and vacuum state is allowed.

We study the Cauchy problem for the equations describing a heat-conducting compressible fluid in three dimensional space with non-negative density. The global existence of unique strong solution is established, in case when the initial energy is assumed to be small. In addition, the decay rates for the solutions and their derivatives are derived.

We study a dual-channel recycling closed-loop supply chain (CLSC) and investigate the royalty strategy involving cost-reducing technique for remanufacturing patented products. Facing information asymmetry and market uncertainty, we address the problem where the patent licensor (manufacturer) and licensee (remanufacturer) simultaneously compete in the sales market and the recycling market. We examine the optimal decisions of a decentralized CLSC (D-CLSC) with the manufacturer being the Stackelberg leader. Numerical examples are used to demonstrate how the patented technology (cost-reducing technique) affects the channel players’ behaviors and how to identify the optimal royalty fee. Based on the theoretical derivation and the numerical outcomes, we find that regardless of the CLSC structure (centralized or decentralized), the take-back prices and the total profits will rise with the increase of savings from the licensed technology. In the D-CLSC, (i) the expected profits of the manufacturer and the remanufacturer as well as the royalty fee will also rise with the savings from the licensed technology. (ii) In addition, the wholesale price, retail price, take-back prices, as well as the royalty fee will rise with the degree of information asymmetry. But the retailer’s expected profit will decline. (iii) Finally, the expected profit of the manufacturer will rise with the demand uncertainty and the return uncertainty. For the remanufacturer, this trend is not obvious. Our research provides guidance to resolve conflicts and intellectual property disputes between the original manufacturer and the remanufacturer of the patented product.

In this paper we study a vanishing pressure process for highly compressible Navier-Stokes equations as the Mach number tends to infinity. We first prove the global existence of weak solutions for the pressureless system in the framework [Li-Xin, arXiv:1504.06826v2], where the weak solutions are established for compressible Navier-Stokes equations with degenerate viscous coefficients. Furthermore, a rate of convergence of the density in $L^{\infty}\left(0,T;L^{2}(\rr)\right)$ is obtained, in case when the velocity corresponds to the gradient of density at initial time.

This article using the spatial econometric analysis technology to build a space from the perspective of geographical feature weighting matrix, spatial correlation of the regional economic growth and integration, to explore the ecological innovation capacity and regional spatial spillover effect of economic growth, and using the spatial regression model decomposition of the partial differential method of actual effect based on 2009-2017 panel data from 29 provinces cities and autonomous regions in China. Based on the data of 29 provinces and cities in China from 2009 to 2017, this paper constructs the spatial weight matrix from the perspective of geographical features, establishes the spatial panel econometric model, and investigates the spatial correlation and clustering of regional economic growth uses spatial econometric analysis technology. We explore the spatial spillover effect of ecological innovation ability and regional economic growth, and uses the partial differential method of spatial regression model to decompose this effect. The results show that, firstly, China’s regional economic development is not balanced, and the economic development of neighboring regions is basically the same, which has a certain degree of spatial agglomeration and significant spatial correlation, and China’s economic growth has significant spatial heterogeneity. Second, the ecological innovation environment has a great spillover effect on the economic growth of neighboring regions, and the human resource level and capital investment have significant positive spillover effect between neighboring regions. The coefficient of ecological innovation output is significantly negative. Thirdly, the regional flows of capital input, labor force, the level of opening up, industrial structure and human capital not only have obvious direct effects, but also have a significant promoting effect on economic growth through the spatial spillover effect. The conclusion of this paper provides policy enlightenment for promoting sustainable economic growth among provinces and regions in China.

The ${q}$ -rung orthopair fuzzy set ( ${q}$ -ROFS) is a powerful tool to deal with uncertainty and ambiguity in real life. The theoretical basis for processing the continuous ${q}$ -rung orthopair fuzzy information is ${q}$ -rung orthopair fuzzy calculus ( ${q}$ -ROFC) and the main object is ${q}$ -rung orthopair fuzzy functions ( ${q}$ -ROFFs). Recently, the authors proposed derivatives and differentials of ${q}$ -ROFFs in the framework of ${q}$ -ROFC. In this paper, we aim to further study the ${q}$ -rung orthopair fuzzy integral ( ${q}$ -ROFI). It is the most important and fundamental part of the ${q}$ -ROFC theoretical system with direct and powerful applications. Our contribution is the indefinite and definite integrals, and bridges the fuzzy calculus theoretical gap of the nonlinear ${q}$ -ROFFs. In particular, we begin with the indefinite integral of ${q}$ -ROFFs, which can be regarded as the anti-derivatives operations of our previous work. Some of their basic properties are discussed. Next, we give the accurate concept of definite integrals of ${q}$ -ROFFs under additive operations, and obtain the explicit integral formula. Some properties of ${q}$ -ROFIs, such as comparison, algebraic operations, and mean value theorem are analyzed. Finally, we generalize the ${q}$ -ROFI to the case when membership and nonmembership functions are allowed to be correlated. After the theoretical results have been established, we present some numerical examples to demonstrate the rationality and effectiveness of integrating continuous ${q}$ -rung orthopair fuzzy data with the ${q}$ -ROFIs.

We study the Cauchy problem for the equations describing a viscous compressible and heat-conductive fluid in two dimensions. By imposing a weight function to initial density to deal with Sobolev embedding in critical space, and constructing an ad-hoc truncation to control the quadratic nonlinearity appeared in energy equation, we establish the local in time existence of unique strong solution with large initial data. The vacuum state at infinity or the compactly supported density is permitted. Moreover, we provide a different approach and slightly improve the weighted \begin{document}$ L^{p} $\end{document} estimates in [ 19 ,Theorem B.1].

This paper is concerned with a diffuse interface model for the flow of two viscous incompressible fluids with matched densities. We establish a L p - L s regularity condition on the gradient of the velocity field such that the weak solution conserves its energy for all positive time.

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.

This paper deals with the initial boundary value problem for one-dimensional (1D) viscous, compressible and heat conducting fluids. We establish the global existence and uniqueness of classical solutions, with large data and possible vacuum at initial time. Our approach is based on the Calder\(\acute{o}\)n–Zygmund decomposition technique and allows that the viscosity and heat conductivity are both constant.

We study the blowup phenomena of solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients in arbitrary dimensions. By constructing a family of self-similar analytical solutions with spherical symmetry, some interesting information including the blowup and expanding properties are shown. In addition, the case of constant viscosity coefficients is also considered. The approach is based on the phase plane method.

This paper deals with the zero dissipation limit problem for the Navier-Stokes equations when the viscosity and the heat-conductivity are of the same order. In the case when the Riemann solution of the Euler equations is piecewise constants with a contact discontinuity, we prove that there exist global solutions to the compressible Navier-Stokes equations, which converge to the in-viscid solution away from the contact discontinuity on any finite time interval, at some convergence rate as the dissipations tend towards zero. In addition, a faster convergence rate is obtained, so long as the strength of contact discontinuity $\delta=|\theta_+ -\theta_-|$ is taken suitably small.

This paper is concerned with the critical exponents for the porous medium equation $u_{t}=\triangle u^m+a(x)u^p, (x,t)\in R^N\times (0,T), $ where $m>1, p>0,$ and the function $a(x)\geq 0$ has a compact support. Suppose the space dimension $N\geq 2$, we prove that the global exponent $p_0$ and the Fujita type exponent $p_c$ are both $m$: if $0 m$ both the blowing up solutions and the global solutions exist. While for the one-dimensional case, it is proved $p_0=\frac{m+1}{2} < m+1 = p_c$ by [E. Ferreira, A. Pablo, J. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Diff. Eqns., 231(2006) 195-211].

This paper deals with the doubly degenerate reaction-di?usion equation where , , and B(0,1) denotes a unit ball in RN with the center in origin. We prove that the blow up phenomenon can be restrained if the space dimension N is taken su?ciently large. Moreover, the critical condition guaranteeing the absence (or occurrence) of the blow up is achieved.

The present paper is concerned with the blow up rate of a solution to the following Cauchy problem u t = Δ u m + u p , ( x , t ) ∈ R N × ( 0 , T ) , with the bounded initial function u ( x , 0 ) = u 0 ( x ) ≥ 0 , where T ∞ is the blow up time. When u 0 ( x ) has a small oscillation around a radial function, we establish the growth rate estimate of the form ‖ u ( ⋅ , t ) ‖ L ∞ ( R N ) ≤ C ( T − t ) − 1 p − 1 , ∀ t ∈ [ 0 , T ) , where the constant C is independent of t -variable, and the parameters m and p are expected to be 1 m p ≤ m ( N + 2 ) ( N − 2 ) + . Although this result is clear, it seems that no rigorous proof can be found up until now.

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.

This paper deals with the large time behavior of nonnegative solutions to the equation $$u_t = div\left( {\left| {\nabla u} \right|^{p - 2} \nabla u} \right) + a\left( x \right)u^q ,\left( {x,t} \right) \in R^N \times (0,T),$$ where p > 2, q > 0, and the function a(x) ≥ 0 has a compact support. We obtain the critical exponent for global existence q0 and the Fujita exponent qc. In one-dimensional case N = 1, we have \(q_0 = \frac{{2(p - 1)}} {p}\) and qc = 2(p − 1). Particularly, all solutions are global in time if 0 qc both blowing up solutions and global solutions exist. However, for the case N ≥ p > 2, these two critical exponents are exactly the same. Namely, q0 = p − 1 = qc.

This paper is concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above independently of both time and space. Moreover, the global solution is showed to be asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.

In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u : = u ( x , t ) be a solution of the Cauchy problem u t = div ( | ∇ u | p − 2 ∇ u ) + u q , u ( x , 0 ) = u 0 ( x ) , where ( x , t ) ∈ R N × ( 0 , T ) , N ⩾ 1 and p > 2 . When q ⩾ p − 1 , we prove that if the initial data u 0 ( x ) has a compact support, then the solution u ( ⋅ , t ) has also compactly support. Moreover, when 1 q p − 1 , we show that the solution of the Cauchy problem blows up at any point of R N to arbitrary compactly supported initial data.