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We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier–Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically symmetric, and away from the vacuum. The solutions obtained here are of global finite total relative-energy including the origin, while cavitation may occur as balls centred at the origin of symmetry for which the interfaces between the fluid and the vacuum must be upper semi-continuous in space-time in the Eulerian coordinates. On any region strictly away from the possible vacuum, the velocity and specific internal energy are Hölder continuous, and the density has a uniform upper bound. To achieve this, our main strategy is to regard the Cauchy problem as the limit of a series of carefully designed initial-boundary value problems that are formulated in finite annular regions. For such approximation problems, we can derive uniform a priori estimates that are independent of both the inner and outer radii of the annuli considered in the spherically symmetric Lagrangian coordinates. The entropy inequality is recovered after taking the limit of the outer radius to infinity by using Mazur’s lemma and the convexity of the entropy function, which is required for the limit of the inner radius tending to zero. Then the global weak solutions of the original problem are attained via careful compactness arguments applied to the approximate solutions in the Eulerian coordinates.

We revisit the moving least squares (MLS) approximation scheme on the sphere (mathbb S^{d-1} subset {mathbb R}^d), where (d>1). It is well known that using the spherical harmonics up to degree (L in {mathbb N}) as ansatz space yields for functions in (mathcal {C}^{L+1}(mathbb S^{d-1})) the approximation order (mathcal {O}left( h^{L+1} right) ), where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as (h rightarrow 0). Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.

We consider a gas of bosons interacting through a hard-sphere potential with radius (mathfrak {a}) in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term (4pi rho mathfrak {a}) and shows that corrections are smaller than (C rho mathfrak {a} (rho {{mathfrak {a}}}^3)^{1/2}), for a sufficiently large constant (C > 0). In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order (rho mathfrak {a}(rho {{mathfrak {a}}}^3)^{1/2}), in agreement with the Lee–Huang–Yang prediction.

When taking a regular planar polygon of (M) sides and length (2pi ) as the initial datum of the vortex filament equation, (mathbf{X}_{t}= mathbf{X}_{s}wedge mathbf{X}_{ss}), the solution becomes polygonal at times of the form (t_{pq} = (p/q)(2pi /M^{2})), with (gcd (p,q)=1), and the corresponding polygon has (Mq) sides, if (q) is odd, and (Mq/2) sides, if (q) is even. Moreover, that polygon is skew (except when (q = 1) or (q = 2), where the initial shape is recovered), and the angle (rho ) between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time (t_{pq}), (cos ^{q}(rho /2) = cos (pi /M)), if (q) is odd, and (cos ^{q}(rho /2) = cos ^{2}(pi /M)), if (q) is even. Since the transition of one side of the polygon to the next one is given by a rotation in (mathbb{R}^{3}) determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle (2pi /M), which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.

In this paper, we consider Cauchy problem for the 2D incompressible Keller-Segel-Navier-Stokes equations with the fractional diffusion

where (wedge :=(-Delta )^{frac{1}{2}}) and (alpha in [frac{1}{2},1]). We get the global well-posedness for the above system with the rough initial data by a new priori estimate of the solutions.