Narrow Results

We consider the steady compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain. We prove the existence of a solution for arbitrarily large data under the assumption that the pressure p(ϱ, θ) ~ ϱθ + ϱ^γ for assuming either the slip or no-slip boundary condition for the velocity and the Newton boundary condition for the temperature. The regularity of solutions is determined by the basic energy estimates, constructed for the system.

We consider a problem modeling the steady flow of a compressible heat conducting Newtonian fluid subject to the slip boundary condition for the velocity. Assuming the pressure law of the form p(ϱ, ϑ) ~ ϱ^γ + ϱϑ, we show (under additional assumptions on the heat conductivity and the viscosity) that for any γ > 1 there exists a variational entropy solution to our problem (i.e. the weak formulation of the total energy balance is replaced by the entropy inequality and the global total energy balance). Moreover, if (together with further restrictions on the heat conductivity), the solution is in fact a weak one. The results are obtained without any restriction on the size of the data.

We are concerned with a system of partial differential equations (PDEs) describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection–diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long-time and large-data existence of weak solutions to the evolutionary problem in three dimensions.

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an L^∞ bound for C¹ solutions of the one-dimensional (1D) Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for 1D magnetohydrodynamics (MHD) with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three-dimensional (3D) flow on the exterior of a ball.

This paper is devoted to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions or, more generally, nonconstant monotonic bounded functions as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multi-type version of the sticky particle dynamics and we obtain the existence of global weak solutions via a compactness argument. We then derive a ${\mathrm{\text{L}}}^p$ stability estimate on the particle system which is uniform in the number of particles. This allows us to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161(1) (2005) 223–342]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all order, which extends the classical ${\mathrm{\text{L}}}^1$ estimate and generalizes to diagonal systems a result by Bolley, Brenier and Loeper [Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ.2(1) (2005) 91–107] in the scalar case. Our results are established without any smallness assumption on the variation of the data, and we only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic.

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