arXiv subjects
Publications and source records attributed to Paolo Maremonti.
In this note we furnish a new result concerning the well known théorème de structure by Leray related to a weak solution to the Navier-Stokes equations. We are able to furnish a new estimate on the left endpoint of the unbounded interval of regularity.
In this paper, we prove the existence and a partial regularity of a weak solution to the system governing the interaction between a rigid body and a viscous incompressible Newtonian fluid. The evolution of the system body-fluid is studied in a frame attached to the body. The choice of this special frame becomes critical from an analytical point of view due to the presence of the term $ω\times x\cdot\nabla u$ in the balance of momentum equation for the fluid. As a consequence, we are forced to look for a technique that is different from the ones usually employed both for the existence and for the partial regularity of a weak solution to the Navier-Stokes problem. Hence, we prove the existence of a weak solution in an original way and give a new proof of the celebrated Théorème de Structure due to Leray. However, the regularity obtained for our weak solution is only for large times, hence our result is weaker compared to the one obtained by Leray.
It is well known that a Leray's weak solution to the Navier-Stokes Cauchy problem enjoys a partial regularity which is known in the literature as the Théorème de Structure of a Leray's weak solution. As well, this result has been extended by some authors to the case of the IBVP. In this note, we achieve the Théorème de Structure by means of a new proof. Our proof is based on a priori estimates for a suitable approximating sequence. In this way our result covers a more general setting in the sense that, e.g., we can also include the case of the weak solutions furnished by Hopf for an IBVP in bounded domains without requiring an energy inequality in a strong form, but just employing a priori estimates on the Galerkin approximation.
It is well known that a Leray-Hopf weak solution enjoys an energy inequality. Here, we investigate the energy equality related to a suitable weak solution to the Navier-Stokes initial boundary value problem. The term suitable is meant in the sense that for our goals we achieve a weak solution whose existence is based as limit of solutions to the mollified Navier-Stokes system. In the case of a weak regularity of the solution, our results justify the possible gap for the energy equality in terms of "kinetic energy". However, if there is a sufficient regularity, e.g., like the continuity of the L2-norm of the weak solution, then the energy equality holds.
In this note, we show two results in the setting of Galdi-Silvestre strong solutions for the rigid body-viscous fluid interaction. The former, under an additional integrability assumption on the gradient of the initial data, proves that the time derivative of the solution belongs to $L^2(0,T;L^2(Ω))$. The latter, thanks to a further assumption only on one solution, proves that the uniqueness holds in the quoted setting. However, our extra assumption for the uniqueness is certainly verified under the integrability assumption on the gradient of the initial data. Hence, the set of solutions enjoying the uniqueness is not empty.
We consider the Navier-Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by G.P. Galdi and P. Maremonti "On the stability of steady-state solutions to the Navier-Stokes equations in the whole space", JMFM, 25 (2023). The authors of the quoted paper consider a spatial study of stability of steady fluid motions. The result hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the nD Navier-Stokes Cauchy problem, n greater than 2. We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.
We give sufficient conditions on the initial data for the decay in time of the kinetic energy, $E$, of solutions to the system of equations describing the motion of a rigid body in a Navier-Stokes liquid. More precisely, assuming the initial data ``small" in appropriate norm, we show that if, in addition, the initial velocity field of the liquid, $v_0$, is in $L^q$, $q\in(1,2)$, then $E(t)$ vanishes as $t\to\infty$ with a specific order of decay. The order remains, however, unspecified if $v_0\in L^2$.
Under the assumption of an initial datum divergence free and in L2, we prove the existence of a weak solution to the Navier-Stokes initial boundary value problem enjoying the energy equality on (0,t), almost everywhere in t>0, in particular, for all t $\in [θ, \infty)$, with $θ:= θ(||v0||2)$. Also, the result allows us to refine some others.
The paper is concerned with the IBVP of the Navier-Stokes equations. The result of the paper is in the wake of analogous results obtained by the authors in previous articles [4, 5]. The goal is to estimaste the possible gap between the energy equality and the energy inequality deduced for a weak solution.
The paper is concerned with the IBVP of the Navier-Stokes equations. The goal is the attempt to construct a weak solution enjoying an energy equality. This result is a natural continuation and improvement of the one obtained by the same authors in [2].
In this note we investigate the existence of time-periodic solutions to the $p$-Navier-Stokes system in the singular case of $p\in (1, 2)$, that describes the flows of an incompressible shear-thinning fluid. In the $3D$ space-periodic setting and for $p \in [ \frac{5}{3} , 2)$ we prove the existence of a regular time-periodic solution corresponding to a time periodic force data which is assumed small in a suitable sense. As a particular case we obtain `regular' steady solutions.
The paper is concerned with estimates of the gradient of the solutions to the Stokes IBVP both in a bounded and in an exterior domain. More precisely, we look for estimates of the kind $\|\nabla v(t)\|_q\leq g(t)\|\nabla v_0\|_p, q\geq p>1$, for all $t>0$, where function $g$ is independent of $v_0$.
We consider the initial boundary value problem for the p(t, x)-Laplacian system in a bounded domain Ω. If the initial data belongs to L^{r_0}, r_0 \geq 2, we give a global L^{r_0}(Ω)-regularity result uniformly in t>0 that, in the particular case r_0 =\infty, implies a maximum modulus theorem. Under the assumption p- = \inf p(t, x) > 2n/(n+r_0), we also state L^{r_0}- L^r estimates for the solution, for r \geq r_0. Complete proofs of the results presented here are given in the paper [F. Crispo, P. Maremonti, M. Ruzicka, Global L^r-estimates and regularizing effect for solutions to the p(t, x) -Laplacian systems, accepted for publication on Advances in Differential Equations, 2017].
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
The paper is concerned with the regularity of weak solutions to the Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin condition in order to obtain regularity for t > 0. The most interesting aspect of the result is that no compatibility condition is required to the initial data $v_0\in J^2(\OO) J2(Ω)$.
Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$. Finn raised the question whether his steady slutions are attainable as limits for $t\to\infty$ of unsteady solutions starting from motionless state when $h(t)=1$ after some finite time and $h(0)=0$ (starting problem). This was affirmatively solved by Galdi, Heywood and Shibata for small $u_\infty$. We study some generalized situation in which unsteady solutions start from large motions being in $L^3$. We then conclude that the steady solutions for small $u_\infty$ are still attainable as limits of evolution of those fluid motions which are found as a sort of weak solutions. The opposite situation, in which $h(t)=0$ after some finite time and $h(0)=1$ (landing problem), is also discussed. In this latter case, the rest state is attainable no matter how large $u_\infty$ is.
Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg, as a corollary, under suitable assumptions of local character on the initial data, we prove a behavior in time of the $L^\infty_{loc}$-norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator. Besides its own interest, the result is a main tool to study the spatial decay estimates of a suitable weak solution, performed in paper F. Crispo and P. Maremonti, On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem (submitted).
We prove space-time decay estimates of suitable weak solutions to the Navier-Stokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result obtained by the authors in the paper "A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem" (submitted), on the behavior of the solution in a neighborhood of $t=0$ in the $L^\infty_{loc}$-norm, which enables us to furnish a representation formula for a suitable weak solution. The second is the asymptotic behavior of the $L^2(\R^3\setminus B_R)$ norm of $u(t)$ for $R\to\infty$. Following a Leray's point of view, roughly speaking our result proves that a possible space-time turbulence does not perturb the asymptotic spatial behavior of the initial data of a suitable weak solution.