Journal of Mathematical Fluid Mechanics最新文献

This work analyzes the finite element approximation to a viscoelastic flow model, which generalizes the Navier–Stokes equation and Oldroyd’s model by introducing the tempered power-law memory kernel. We prove regularity and long-time exponential decay of the solutions, as well as a long-time convolution-type Grönwall inequality to support numerical analysis. A Volterra–Stokes projection is developed and analyzed to facilitate the parabolic-type duality argument, leading to the long-time error estimates and exponential decay of velocity and pressure. A benchmark problem of planar four-to-one contraction flow is simulated to substantiate the generality of the proposed model in comparison with the Navier–Stokes equation and Oldroyd’s model.

This paper is concerned with the time-decay rates of the viscous contact wave in 1D heat-conductive ideal gas without viscosity. For the case without the zero mass condition, although the heat-conductive ideal gas without viscosity studied in this paper lacks dissipation in terms of velocity, we can still prove the same optimal decay rate as the compressible Navier–Stokes equation. Precisely, we can obtain the optimal decay rate ((1+t)^{-1/2}) for the perturbation in (L^{infty })-norm. The crucial point of the analysis is as follows. We refine the estimates for both anti-derivatives and the original perturbations in the first, and then, we use an innovative transformation to ensure the structural conditions continue to hold for the system of derivatives. With this approach, we obtain the optimal decay rate.

We consider steady water waves in a two-dimensional channel bounded below by a flat, rigid bottom and above by a free surface. Surface tension is neglected, and the flow is rotational with constant vorticity a. We analyze an analytic branch of Stokes waves bifurcating from a subcritical laminar flow, with the wave period serving as the bifurcation parameter. Along this branch, the first eigenvalue of the Fréchet derivative remains negative. Our main focus is the second eigenvalue; its sign plays a crucial role in the analysis of subharmonic bifurcations. This small eigenvalue determines the validity of the principle of exchange of stabilities: a positive sign confirms it, while a negative sign indicates its violation. Furthermore, a positive second eigenvalue corresponds to an increasing period along the bifurcation curve near the critical point, whereas a negative sign implies period decrease. We investigate how the sign of the second eigenvalue depends on the Bernoulli constant R (equivalently, the laminar flow depth d) and the vorticity a. We show that for each a there exists a critical depth (d_0(a)) such that the second eigenvalue is positive for (d<d_0(a)) and negative for (d>d_0(a)). In the laminar flow, a stagnation point forms when the depth exceeds a threshold (d_{text {s}}(a)). We demonstrate that (d_0(a) < d_{text {s}}(a)) for (a > a_0 approx -1.01803), whereas (d_0(a) > d_{text {s}}(a)) for (a < a_0). We also verify the property of formal stability by a description of the domain in (a, d) variables, where this property holds. Numerical illustrations of these properties are presented in the paper.

We consider a model of the compressible non-Newtonian fluids for power-law flow fulfilling a periodic domain in ({mathbb R}^3,) in which the extra stress tensor is induced by a potential with p(t, x)-structure. The local-in-time existence and uniqueness of strong solutions is proved for all (frac{7}{5} < inf p(t,x) leqslant sup p(t,x) leqslant 2.) Further, two improved blow-up criterions for the obtained strong solution is given.

We study the three-dimensional hyperdissipative Navier–Stokes system in the super-critical regime (below the Lions threshold). Leveraging the analyticity–sparseness gap in the case where the ratios of the lower-order derivatives with respect to the higher-order derivatives are bounded by constants, we introduce time-weighted bridge inequalities quantifying these ratios across all derivative levels. On one hand, the time weights are chosen in a way that the resulting lower bound on the radius of spatial analyticity still dominates the scale of sparseness of the super-level sets indicating that the flow entered the dissipation regime. On the other hand, they provide a less restrictive environment for the ratios giving the construction more flexibility. As an illustration of its utility, it is shown that – together with a harmonic-measure contraction on one-dimensional sparse sets – this mechanism rules out a more general class of the blow-up scenarios compared to the previous work by Farhat and Grujić (essentially a perturbation of the multidimensional geometric series).

We establish the existence of weak solutions for a class of fully coupled multilayered fluid–structure interaction systems in a three-dimensional spatial setting. The model consists of an incompressible viscous fluid interacting with a thin elastic shell, which is in turn coupled to a three-dimensional elastic solid, yielding a nonstandard 3D/2D/3D coupling configuration. The system is driven by time-periodic boundary forcing through Bernoulli-type pressure conditions. For sufficiently small forcing in (L^2), we prove the existence of at least one time-periodic weak solution. A central analytical difficulty stems from the strong nonlinear coupling across interfaces of different dimensionality and the absence of classical compactness mechanisms. This challenge is overcome through a carefully designed energy framework combined with and new (L^{2}) compactness arguments adapted to the multilayered geometry. A key structural assumption is the viscoelasticity of the three-dimensional solid, which yields additional diffusion estimates and ensures energy stability. In the purely elastic case, we establish the global-in-time existence of weak solutions to the corresponding initial-value problem, provided that no degeneration (self-contact) of the fluid domain occurs. These results extend existing theories for two-dimensional and reduced-dimensional configurations to a genuinely three-dimensional multilayered setting, providing new analytical insight into complex coupled PDE systems arising in fluid–structure interaction.

In the present paper, we investigate the existence, uniqueness, and localization of solutions for a recently derived model for gyres, formulated as a Dirichlet problem. This problem is studied from two perspectives. In the first part, we establish the existence and uniqueness of a weak solution by means of the Minty–Browder theorem. For the nonlinear term, we assume a suitable monotonicity condition together with an exponential growth condition. Here, the growth condition is admissible since the problem is posed in dimension two, which allows us to use the Trudinger–Moser inequality. In the second part, we prove the existence of a localized positive solution within a conical annular region by means of the Moser–Harnack inequality. This approach also yields multiplicity results when the required conditions are satisfied on disjoint sets. From a physical perspective, an upper bound for the solution reflects the dynamical intensity of the flow, while a lower bound indicates the persistence of ocean circulation in a given region. These bounds are determined by the behavior of the vorticity function.

We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and Itô noise are treated.

It is now well known that the classical notion of admissible weak solutions (also known as weak entropy solutions) does not restore uniqueness for the multi-dimensional compressible Euler equations. Indeed, convex integration has shown that admissible weak solutions are in general highly non-unique. This has motivated additional selection criteria intended to rule out the counterintuitive solutions generated by convex integration. In this paper we prove that the local least action criterion introduced by H. Gimperlein, M. Grinfeld, R. J. Knops and M. Slemrod does not serve as a proper selection criterion, since it fails to select the solution that is intuitively expected to be physically relevant.

In this paper, we construct a new class of uniformly rotating vortex sheets arising as perturbations of the circular configuration within the Kelvin–Helmholtz model for two incompressible fluids of equal density, with surface tension effects taken into account. Despite the fundamental importance of the problem, few results are available on the long-time dynamics, due to the strong instability inherent in the Kelvin–Helmholtz mechanism. We carry out a detailed bifurcation analysis and construct families of nontrivial rotating solutions that bifurcate from the circular state when either the rotational speed or the surface tension coefficient is varied. The stationary branches obtained by varying the mean vorticity follow from the same analysis through the scaling symmetry of the equations.