arXiv subjects
Publications and source records attributed to Julien Guillod.
We construct self-similar solutions to the 2D Navier--Stokes equations evolving from arbitrarily large $-1$--homogeneous initial data and present numerical evidence for their non-uniqueness.
Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.
By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.
We consider here a two-dimensional incompressible fluid in a periodic channel, whose density is advected by pure transport, and whose velocity is given by the Stokes equation with gravity source term. Dirichlet boundary conditions are taken for the velocity field on the bottom and top of the channel, and periodic conditions in the horizontal variable. We prove that the affine stratified density profile is stable under small perturbations in Sobolev spaces and prove convergence of the density to another limiting stratified density profile for large time with an explicit algebraic decay rate. Moreover, we are able to precisely identify the limiting profile as the decreasing vertical rearrangement of the initial density. Finally, we show that boundary layers are formed for large times in the vicinity of the upper and lower boundaries. These boundary layers, which had not been identified in previous works, are given by a self-similar Ansatz and driven by a linear mechanism. This allows us to precisely characterize the long-time behavior beyond the constant limiting profile and reach more optimal decay rates.
We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special symmetry and smallness assumptions. In the paper we solve the key difficulties in applying Leray's {\it invading domains method} and, as a consequence, prove the existence of $D$-solutions in the whole plane for arbitrary compactly supported force. The boundary condition at infinity are verified in two different scenarios: (I) the~limiting velocity is sufficient large with respect to the external force, (II) both the total integral of force and the~limiting velocity vanish. Hence, our method produces large class of new solutions with prescribed spatial limits. Moreover, we show the uniqueness of $D$-solutions to this problem in a perturbative regime. The main tools here are two new estimates for general Navier-Stokes solutions, which have rather simple forms. They control the difference between mean values of the velocity over two concentric circles in terms of the Dirichlet integral in the annulus between them.
The real Ginzburg-Landau equation possesses a family of spatially periodic equilibria. If the wave number of an equilibrium is strictly below the so called Eckhaus boundary the equilibrium is known to be spectrally and diffusively stable, i.e., stable w.r.t. small spatially localized perturbations. If the wave number is above the Eckhaus boundary the equilibrium is unstable. Exactly at the boundary spectral stability holds. The purpose of the present paper is to establish the diffusive stability of these equilibria. The limit profile is determined by a nonlinear equation since a nonlinear term turns out to be marginal w.r.t. the linearized dynamics.
We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$ for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the reflection with respect to the $xy$-plane. Working in the class of axi-symmetric fields, we calculate numerically scale-invariant solutions of the Cauchy problem in terms of their profile functions, which are smooth. The solutions are necessarily unique for small data, but for large data we observe a breaking of the reflection symmetry of the initial data through a pitchfork-type bifurcation. By a variation of previous results by Jia & \v{S}ver\'ak (2013) it is known rigorously that if the behavior seen here numerically can be proved, optimal non-uniqueness examples for the Cauchy problem can be established, and two different solutions can exists for the same initial datum which is divergence-free, smooth away from the origin, compactly supported, and locally $(-1)$-homogeneous near the origin. In particular, assuming our (finite-dimensional) numerics represents faithfully the behavior of the full (infinite-dimensional) system, the problem of uniqueness of the Leray-Hopf solutions (with non-smooth initial data) has a negative answer and, in addition, the perturbative arguments such those by Kato (1984) and Koch & Tataru (2001), or the weak-strong uniqueness results by Leray, Prodi, Serrin, Ladyzhenskaya and others, already give essentially optimal results. There are no singularities involved in the numerics, as we work only with smooth profile functions. It is conceivable that our calculations could be upgraded to a computer-assisted proof, although this would involve a substantial amount of additional work and calculations, including a much more detailed analysis of the asymptotic expansions of the solutions at large distances.
The existence of weak solutions to the stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ is proven. This particular geometry was the only case left open since the work of Leray in 1933. The reason is that due to the absence of boundaries the local behavior of the solutions cannot be controlled by the enstrophy in two dimensions. We overcome this difficulty by constructing approximate weak solutions having a prescribed mean velocity on some given bounded set. As a corollary, we obtain infinitely many weak solutions in $\mathbb{R}^2$ parameterized by this mean velocity, which is reminiscent of the expected convergence of the velocity field at large distances to any prescribed constant vector field. This explicit parameterization of the weak solutions allows us to prove a weak-strong uniqueness theorem for small data. The question of the asymptotic behavior of the weak solutions remains however open, when the uniqueness theorem doesn't apply.
The Navier-Stokes equations in a two-dimensional exterior domain are considered. The asymptotic stability of stationary solutions satisfying a general hypothesis is proven under any $L^2$-perturbation. In particular the general hypothesis is valid if the steady solution is the sum of the critically decaying flux carrier with flux $|Φ| < 2π$ and a small subcritically decaying term. Under the central symmetry assumption, the general hypothesis is also proven for any critically decaying steady solutions under a suitable smallness condition.
This study is devoted to the incompressible and stationary Navier-Stokes equations in two-dimensional unbounded domains. First, the main results on the construction of the weak solutions and on their asymptotic behavior are reviewed and structured so that all the cases can be treated in one concise way. Most of the open problems are linked with the case of a vanishing velocity field at infinity and this will be the main subject of the remainder of this study. The linearization of the Navier-Stokes around the zero solution leads to the Stokes equations which are ill-posed in two dimensions. It is the well-known Stokes paradox which states that if the net force is nonzero, the solution of the Stokes equations will grow at infinity. By studying the link between the Stokes and Navier-Stokes equations, it is proven that even if the net force vanishes, the velocity and pressure fields of the Navier-Stokes equations cannot be asymptotic to those of the Stokes equations. However, the velocity field can be in some cases asymptotic to two exact solutions of the Stokes equations which also solve the Navier-Stokes equations. Finally, a formal asymptotic expansion at infinity for the solutions of the two-dimensional Navier-Stokes equations having a nonzero net force is established based physical arguments. The leading term of the velocity field in this expansion decays like $|\boldsymbol{x}|^{-1/3}$ and exhibits a wake behavior. Numerical simulations are performed to validate this asymptotic expansion when is net force is nonzero and to analyze the asymptotic behavior in the case where the net force is vanishing. This indicates that the Navier-Stokes equations admit solutions whose velocity field goes to zero at infinity in contrast to the Stokes linearization and moreover this shows that the set of possible asymptotes is very rich.
The asymptotic behavior of the vorticity for the steady incompressible Navier-Stokes equations in a two-dimensional exterior domain is described in the case where the velocity at infinity $\boldsymbol{u}_{\infty}$ is nonzero. It is well known that the asymptotic behavior of the velocity field is given by the fundamental solution of the Oseen system which is the linearization of the Navier-Stokes equation around $\boldsymbol{u}_{\infty}$. The vorticity has the property of decaying algebraically inside a parabolic region called the wake and exponentially outside. The previously proven asymptotic expansions of the vorticity are relevant only inside the wake because everywhere else the remainder is larger than the asymptotic term. Here we present an asymptotic expansion that removes this weakness. Surprisingly, the found asymptotic term is not given by the Oseen linearization and has a power of decay that depends on the data. This strange behavior is specific to the two dimensional problem and is not present in three dimensions.
We consider the stationary incompressible Navier-Stokes equation in the half-plane with inhomogeneous boundary condition. We prove existence of strong solutions for boundary data close to any Jeffery-Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery-Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove existence of weak solutions, as well as weak-strong uniqueness for small data.
New explicit solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^{2}\setminus\left\{ \boldsymbol{0}\right\}$ are determined, which generalize the scale-invariant solutions found by Hamel. These new solutions are invariant under a particular combination of the scaling and rotational symmetries. They are the only solutions invariant under this new symmetry in the same way as the Hamel solutions are the only scale-invariant solutions. While the Hamel solutions are parameterized by a discrete parameter $n$, the flux $Φ$ and an angle $θ_{0}$, the new solutions generalize the Hamel solutions by introducing an additional parameter $a$ which produces a rotation. The new solutions decay like $\left|\boldsymbol{x}\right|^{-1}$ as the Hamel solutions, and exhibit spiral behavior. The new variety of asymptotes induced by the existence of these solutions further emphasizes the difficulties faced when trying to establish the asymptotic behavior of the Navier-Stokes equations in a two-dimensional exterior domain or in the whole plane.
We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier-Stokes equations in a two-dimensional exterior domain. More precisely, we find the asymptotic behaviour for such solutions in the case where the net force on the boundary of the domain is non-zero. In contrast to the three dimensional case, where the asymptotic behaviour is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width $|\boldsymbol{x}|^{2/3}$, the velocity decays like $|\boldsymbol{x}|^{-1/3}$, whereas outside the wake, it decays like $|\boldsymbol{x}|^{-2/3}$. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the non-zero and the zero net force case.