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Abstract: Abstract In this paper, the one-dimensional compressible Navier-Stokes system with outer pressure boundary conditions is investigated. Under some suitable assumptions, we prove that the specific volume and the temperature are bounded from below and above independently of time, and then give the local and global existence of strong solutions. Furthermore, we also obtain the convergence of the global strong solution to a stationary state and the nonlinear stability of its convergence. It is worth noticing that all the assumptions imposed on the initial data are the same as Takeyuki Nagasawa [Japan.J.Appl.Math.(1988)]. Therefore, our work can be regarded as an improvement of the results of Nagasawa. PubDate: 2022-05-17
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Abstract: Abstract In this paper, we investigate the large-time behavior for the non-isentropic compressible Navier–Stokes equations with capillarity in the whole space \({\mathbb {R}}^{d}\) ( \(d\ge 3\) ). Under an additional smallness assumption of the low frequencies of initial data, the time-decay estimates of \(L^{q}\) – \(L^{r}\) type for global strong solutions near constant equilibrium (away from vacuum) can be deduced by establishing the time-weighted energy inequality. On the other hand, a pure energy approach (without the spectral analysis) different from the time-weighted energy method is performed, which allows us not only to get the time-decay rates but also to remove the smallness condition of low frequencies of initial data. The treatment of new nonlinear terms arising from capillary mainly depends on non classical Besov product estimates and the refined use of Sobolev embeddings and interpolations. PubDate: 2022-05-12
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Abstract: Abstract This paper is concerned with the global well-posedness and diffusion limit (as \(\varepsilon \rightarrow 0\) ) of radial solutions to a chemotaxis system with logarithmic singular sensitivity in a bounded interval with mixed Dirichlet and Robin boundary conditions. We use a Cole–Hopf type transformation to resolve the logarithmic singularity and prove the global well-posedness of the transformed system with \(\varepsilon \) equaling to 0 or being suitably small. Moreover, the transformed system is justified to possess boundary layer effects as \(\varepsilon \rightarrow 0\) , where the boundary layer thickness is of \(\mathcal {O}(\varepsilon ^{\alpha })\) with \(0<\alpha <\frac{1}{2}\) . By transferring the results back to the original chemotaxis model via Cole–Hopf transformation, we find that boundary layer profile is present at the gradient of solutions and the solution itself is uniformly convergent with respect to \(\varepsilon >0\) . PubDate: 2022-05-06
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Abstract: Abstract This paper is devoted to studying the existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation without any additional assumptions. Firstly, introducing a new set of variables, we transform the single-cycle pulse equation into an equivalent semilinear system. Using the standard ordinary differential equation theory, the global solution of the semilinear system is studied. Secondly, returning to the original coordinates, we get a global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, choosing some vital test functions which are different from [Bressan (Discrete Contin. Dyn. Syst 35:25-42, 2015), Brunelli (Phys. Lett. A 353:475-478, 2006)], we find a equation to single out a unique characteristic curve through each initial point. Moreover, the uniqueness of global admissible conservative weak solution is obtained. PubDate: 2022-05-04
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Abstract: Abstract In this work we will focus on the existence of dissipative solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. Moreover, we will study under which conditions it would be possible to get the existence of weak solutions. PubDate: 2022-04-27
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Abstract: Abstract Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. Existence theories for fully localised three-dimensional solitary waves on water of finite depth have recently been published, and in this paper we establish their existence on deep water. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations. PubDate: 2022-04-22
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Abstract: Abstract This work is concerned with the global solutions to the d-dimensional incompressible Oldroyd-B model with only dissipation in the equation of stress tensor (without stress tensor damping or velocity dissipation). The main ingredients of the proof lies in commutator estimate at low frequency and energy estimate in Lagrangian coordinates at high frequency. Particularly, our result extends the works of Wu–Zhao [24] (J. Differ. Equ. 316, 2021) and Constantin–Wu–Zhao–Zhu [9] (J. Evol. Equ. 21, 2021). PubDate: 2022-04-22
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Abstract: Abstract We investigate the local interior regularity condition of a suitable weak solution to 3D MHD equations. We prove that if the gradient of a velocity vector belong to a local BMO space \(\mathrm{bmo}_r\) in a neighborhood of an interior point, then solution is regular near that point. PubDate: 2022-04-21
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Abstract: Abstract Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem. PubDate: 2022-04-15
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Abstract: Abstract We consider the forced Nernst–Planck–Navier–Stokes system for n ionic species with different diffusivities and valences. We prove the local existence of analytic solutions with periodic boundary conditions in two and three dimensions. In the case of two spatial dimensions, the local solution extends uniquely and remains analytic on any time interval [0, T]. In the three dimensional case, we give necessary and sufficient conditions for the global in time existence of analytic solutions. These conditions involve quantitatively only low regularity norms of the fluid velocity and concentrations. PubDate: 2022-04-13
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Abstract: Abstract We consider here the simplified Ericksen–Leslie system on the whole space \(\mathbb {R}^{3}\) . This system deals with the incompressible Navier–Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for nematic liquid crystals. For both, the stationary (time independent) case and the non-stationary (time dependent) case, using the fairly general framework of a kind of local Morrey spaces, we obtain some a priori conditions on the unknowns of this coupled system to prove that they vanish identically. This results are known as Liouville-type theorems. As a bi-product, our theorems also improve some well-known results on Liouville-type theorems for the particular case of classical Navier–Stokes equations. PubDate: 2022-04-07
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Abstract: Abstract In the present study several integrable equations with cubic nonlinearity are derived as asymptotic models from the classical shallow water theory. The starting point in our derivation is the Euler equation for an incompressible fluid with the simplest bottom and surface conditions. The approximate equations are obtained by working under suitable scalings that allow for the modeling of water waves of relatively large amplitude, truncating the asymptotic expansions of the unknowns to appropriate order, and introducing a special Kodama transformation. The so obtained equations exhibit cubic order nonlinearities and can be related to the following integrable systems: the Novikov equation, the modified Camassa–Holm equation, and a Camassa–Holm type equation with cubic nonlinearity. Analytically, the formation of singularities of the solution to some of these quasi-linear model equations is also investigated, with an emphasis on the understanding of the effect of the nonlocal higher order nonlinearities. In particular it is shown that one of the models accommodates the phenomenon of curvature blow-up. PubDate: 2022-04-01
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Abstract: Abstract The present paper aims at extending to the Euler–Riesz system the results obtained in (J Hyperbolic Differ Equ 18(1):169–193, 2021; J Evol Equ 21(3):3035–3054, 2021) for the Euler–Poisson and the Euler–Helmholtz systems supplemented with the isentropic pressure law \(\Pi (\varrho )=A\varrho ^\gamma ,\) in the whole space \({\mathbb {R}}^d\) with \(d\ge 2\) . Our first result is the local existence of classical solutions for the associated Cauchy problem in a simple functional framework that does not require the velocity to decay at infinity and the density to be compactly supported. This result is also valid for ideal gases in the isothermal case (that is \(\gamma =1\) ). Next, following the work by Grassin and Serre dedicated to the compressible Euler system (Grassin and Serre in C R Acad Sci Paris Sér I 325:721–726, 1997; Grassin in Indiana Univ Math J 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field \(u_0\) such that the spectrum of \(Du_0\) is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates (the constructed solution is only ‘almost global’ if \(\gamma =1\) ). Our functional framework enables us to address the instability issue of non trivial static solutions in the attractive case for \(\gamma =2d/(d+\sigma ),\) where \(\sigma \) stands for the exponent of the Riesz potential under consideration. PubDate: 2022-03-30
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Abstract: Abstract We establish the validity of the Euler \(+\) Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with the Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an \(L^{1}\) -type norm for the vorticity of the error term in the expansion Navier-Stokes \(-(\) Euler \(+\) Prandtl). An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest PubDate: 2022-03-28
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Abstract: Abstract In Pitaevskii (Sov Phys JETP 35(8):282–287, 1959), a micro-scale model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations (NSE). In this article, we show the local existence of solutions—strong in wavefunction and velocity, weak in density—to this system in a smooth bounded domain in 3D, by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in Kim (SIAM J Math Anal 18(1):89–96, 1987) for the incompressible, inhomogeneous NSE.) To the best of our knowledge, this is the first rigorous mathematical analysis of a bidirectionally coupled system of the NLS and NSE. PubDate: 2022-03-28
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Abstract: Abstract We consider the homogenization of the compressible Navier-Stokes-Fourier equations in a randomly perforated domain in \({\mathbb {R}}^3\) . Assuming that the particle size scales like \(\varepsilon ^\alpha \) , where \(\varepsilon >0\) is their mutual distance and \(\alpha >3\) , we show that in the limit \(\varepsilon \rightarrow 0\) , the velocity, density, and temperature converge to a solution of the same system. We follow the methods of Lu and Pokorný [https://doi.org/10.1016/j.jde.2020.10.032] and Pokorný and Skříšovský [https://doi.org/10.1007/s41808-021-00124-x] where they considered the full system in periodically perforated domains. PubDate: 2022-03-26
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Abstract: Abstract We focus on the Navier–Stokes equations for a compressible viscous fluid—allowing variations of the dynamic eddy viscosity only in the vertical direction—and the continuity equation. Our problem is written in spherical coordinates, in a non-inertial rotating frame. For zonal flows, with no variations in the longitudinal direction, and in a neighbourhood of the Equator, we get a linear parabolic evolution equation that we solve by the method of separation of variables. For a dynamic eddy viscosity which decreases with height above the ground level, the velocity field obtained has an azimuthal component which depends on time and has a nonlinear dependence on the radial coordinate, and a vertical component which depends linearly on the radial coordinate. PubDate: 2022-03-21
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Abstract: Abstract Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains. We present a purely Lagrangian proof that the solution map is strongly continuous in \(L^p\) for all \(p\in [1, \infty )\) and is weakly- \(*\) continuous in \(L^\infty \) . PubDate: 2022-03-21
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Abstract: Abstract Turbulent compressible flows are encountered in many industrial applications, for instance when dealing with combustion or aerodynamics. This paper is dedicated to the study of a simple turbulent model for compressible flows. It is based on the Euler system with an energy equation and turbulence is accounted for with the help of an algebraic closure that impacts the thermodynamical behavior. Thereby, no additional PDE is introduced in the Euler system. First, a detailed study of the model is proposed: hyperbolicity, structure of the waves, nature of the fields, existence and uniqueness of the Riemann problem. Then, numerical simulations are proposed on the basis of existing finite-volume schemes. These simulations allow to perform verification test cases and more realistic explosion-like test cases with regards to the turbulence level. PubDate: 2022-03-21
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Abstract: Abstract The paper deals with the stability of a uniformly rotating finite mass consisting of two immiscible viscous incompressible fluids with unknown interface and exterior free boundary. Capillary forces act on both surfaces. The proof of stability is based on the analysis of an evolutionary problem for small perturbations of the equilibrium state of a rotating two-phase fluid. It is proved that for small initial data and small angular velocity, as well as the positivity of the second variation of energy functional, the perturbation of the axisymmetric equilibrium figure exponentially tends to zero as \(t\rightarrow \infty \) , the motion of the drop going over to the rotation of the liquid mass as a rigid body. PubDate: 2022-03-21
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