In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution

The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier–Stokes equations in the vicinity of an unstable equilibrium solution, by means of a ‘minimal’ and ‘least’ invasive feedback strategy which consists of a control pair \(\{ v,u \}\) (Lasiecka and Triggiani in Nonlinear Anal 121:424–446, 2015). Here v is

We give two characterizations, one for the class of generalized Young measures generated by \({{\,\mathrm{{\mathcal {A}}}\,}}\)-free measures and one for the class generated by \({\mathcal {B}}\)-gradient measures \({\mathcal {B}}u\). Here, \({{\,\mathrm{{\mathcal {A}}}\,}}\) and \({\mathcal {B}}\) are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property

We present results regarding the existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially, in the different proofs of the paper. We also make a link with the trajectories induced by the solution of

We study the asymptotic spatial behavior of the vorticity field, \(\omega (x,t)\), associated to a time-periodic Navier–Stokes flow past a body, \({\mathscr {B}}\), in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, \({\mathcal {R}}\), \(\omega \) decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by \({\bar{\omega }}\)

We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15

We introduce a new geometric–analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with a negative sign. This is motivated by searching quantitative inequalities for the best constants of Sobolev–Poincaré inequalities with trace terms in \({\mathbb {R}}^n\) which correspond to fundamental eigenvalues

We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg–De Vries (gKdV) and Benjamin–Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from almost any (in probability and in density) small and smooth initial data. In particular, we prove a long-time stability result in Sobolev norm: given a large constant

In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion

Let \(\Omega \subset \mathbb {R}^3\) be a Lipschitz domain and let \(S_\mathrm {curl}(\Omega )\) be the largest constant such that $$\begin{aligned} \int _{\mathbb {R}^3}|\nabla \times u|^2\, \mathrm{d}x\ge S_{\mathrm {curl}}(\Omega ) \inf _{\begin{array}{c} w\in W_0^6(\mathrm {curl};\mathbb {R}^3)\\ \nabla \times w=0 \end{array}}\Big (\int _{\mathbb {R}^3}|u+w|^6\,\mathrm{d}x\Big )^{\frac{1}{3}} \end{aligned}$$

It is conjectured that the generalization of the Constantin–Lax–Majda model (gCLM) \(\omega _t + a u\omega _x = u_x \omega \), due to Okamoto, Sakajo and Wunsch, can develop a finite time singularity from smooth initial data for \(a < 1\). For the endpoint case where a is close to and less than 1, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial

We consider two-dimensional (2D) pseudosteady flows around a sharp corner. This problem can be seen as a 2D Riemann initial and boundary value problem (IBVP) for the compressible Euler system. The initial state is a combination of a uniform flow in one quadrant and vacuum in the remaining domain. The boundary condition on the wall of the sharp corner is a slip boundary condition. By a self-similar

Consider a two-dimensional laminar flow between two plates, so that \((x_1,x_2)\in {{\mathbb {R}}}\times [-1,1]\), given by \({{\mathbf {v}}}(x_1,x_2)=(U(x_2),0)\), where \(U\in C^4([-1,1])\) satisfies \(U^\prime \ne 0\) in \([-1,1]\). We prove that the flow is linearly stable in the large Reynolds number limit, in two different cases: \(\sup _{x\in [-1,1]} |U''(x)| + \sup _{x\in [-1,1]} |U'''(x)|

We apply Gamma calculus to the hypoelliptic and non-symmetric setting of Langevin dynamics under general conditions on the potential. This extension allows us to provide explicit estimates on the convergence rate (which is exponential) to equilibrium for the dynamics in a weighted \(H^1(\mu )\) sense, \(\mu \) denoting the unique invariant probability measure of the system. The general result holds

We investigate the phase-field approximation of the Willmore flow by rigorously justifying its sharp interface limit. This is a fourth-order diffusion equation with a parameter \(\varepsilon >0\) that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as \(\varepsilon \) tends to zero the level-set of the solution will converge to the

We prove a \(\varGamma \)-convergence result for a class of Ginzburg–Landau type functionals with \({\mathscr {N}}\)-well potentials, where \({\mathscr {N}}\) is a closed and \((k-2)\)-connected submanifold of \({\mathbb {R}}^m\), in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet

In this work we consider the Landau–de Gennes model for liquid crystals with an external magnetic field to model the occurrence of the Saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous

We analyze generic sequences for which the geometrically linear energy $$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{aligned}$$ remains bounded in the limit \(\eta \rightarrow 0\). Here \( e(u) \,{:}{=}\,1/2(Du + Du^T)\)

A Correction to this paper has been published: https://10.1007/s00205-021-01632-9

We rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms

Global solutions to the compressible Euler equations with heat transport by convection in the whole space are shown to exist through perturbations of Dyson’s isothermal affine solutions (Dyson in J Math Mech 18(1):91–101, 1968). This setting presents new difficulties because of the vacuum at infinity behavior of the density. In particular, the perturbation of isothermal motion introduces a Gaussian

We study the asymptotic behavior, when \(\varepsilon \rightarrow 0\), of the minimizers \(\{u_\varepsilon \}_{\varepsilon >0}\) for the energy $$\begin{aligned} E_\varepsilon (u)=\int _{\Omega }\Big (|\nabla u|^2+\big (\frac{1}{\varepsilon ^2}-1\big )|\nabla |u||^2\Big ), \end{aligned}$$ over the class of maps \(u\in H^1(\Omega ,{{\mathbb {R}}}^2)\) satisfying the boundary condition \(u=g\) on \(\partial

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior

We give a qualitative description of extremals for Morrey’s inequality. Our theory is based on exploiting the invariances of this inequality, studying the equation satisfied by extremals, and the observation that extremals are optimal for a related convex minimization problem.

This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.

We consider a coupled reaction–advection–diffusion system based on the Fisher-KPP and Burgers equations. These equations serve as a one-dimensional version of a model for a reacting fluid in which the arising density differences induce a buoyancy force advecting the fluid. We study front propagation in this system through the lens of traveling waves solutions. We are able to show two quite different

For any compact connected manifold M, we consider the generalized contact Hamiltonian H(x, p, u) defined on \(T^*M\times \mathbb {R}\) which is convex in p and monotonically increasing in u. Let \(u_\varepsilon ^-:M\rightarrow \mathbb {R}\) be the viscosity solution of the parametrized contact Hamilton–Jacobi equation $$\begin{aligned} H(x,d_x u_\varepsilon ^-(x),\varepsilon u_\varepsilon ^-(x))=c(H)

We are concerned with the global weak continuity of the Cartan structural system—or equivalently, the Gauss–Codazzi–Ricci system—on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2), extending the classical quadratic theorem of

By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov–Poisson–Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one

We construct an invariant measure \(\mu \) for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of \(\mu \) are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of \(\mu \) is infinite dimensional, meaning that it is not locally a subset of any compact set with finite

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical

We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein–Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein–Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift \(\kappa >0\). We prove their

We identify a class maximal dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times.

We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier–Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component

The density functional theory (DFT) is a remarkably successful theory of electronic structure of matter. At the foundation of this theory lies the Kohn–Sham (KS) equation. In this paper, we describe the long-time behaviour of the time-dependent KS equation. Assuming weak self-interactions, we prove global existence and scattering in (almost) the full “short-range” regime. This is achieved with new

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier–Stokes equations in dimension three. We show that it is locally \(L ^{\frac{4}{3}, q}\) for any \(q > \frac{4}{3}\), which improves from the current result of \(L ^{\frac{4}{3}, \infty }\). Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions

We consider weak solutions of the Novikov equation that lie in the energy space \(H^1\) with non-negative momentum densities. We prove that a special family of such weak solutions, namely the peakons, is \(H^1\)-asymptotically stable. Such a result is based on a rigidity property of the Novikov solutions which are \(H^1\)-localized and the corresponding momentum densities are localized to the right

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic study of the viscous layer equations and the \(L^2\) to \(L^\infty \) framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations

The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the

We study the nonlinear Schrödinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence \(N_j\) tending to infinity in the whole range of possible p’s, in dimensions \(d\ge 1\). This allows us to prove that translational symmetry is broken for a quantum crystal in the

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These polynomial and exponential \(L^\infty \) bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator. First, we prove a potential type estimate

In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive a sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with

This paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for \(n\times n\) hyperbolic conservation laws in one space dimension. These estimates are achieved by a “post-processing algorithm”, checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number \(\varepsilon >0\) is vanishing. Precisely, whenever \(\varepsilon >0\) is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter \(a\in [\frac{2}{3},1]\), we construct

We construct forward self-similar solutions (expanders) for the compressible Navier–Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity due to cavitation at the origin.

The relativistic quantum Boltzmann equation (or the relativistic Uehling–Uhlenbeck equation) describes the dynamics of single-species fast-moving quantum particles. With the recent development of relativistic quantum mechanics, the relativistic quantum Boltzmann equation has been widely used in physics and engineering, for example in the quantum collision experiments and the simulations of electrons

In this paper we prove a conjecture by P.-L. Lions on maximal regularity of \(L^q\)-type for periodic solutions to \(-\Delta u + |Du|^\gamma = f\) in \(\mathbb {R}^d\), under the (sharp) assumption that \(q > d \frac{\gamma -1}{\gamma }\).

We present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schrödinger type. For Hamiltonian dispersive equations with a non-singular symplectic form and d conserved quantities (in addition to the Hamiltonian), one expects that generically \({{\mathcal {L}}}\), the linearization around a periodic traveling wave, will have a particular Jordan

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour

We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary

We study the exterior problem for stationary Navier–Stokes equations in two dimensions describing a viscous incompressible fluid flowing past an obstacle. It is shown that, at small Reynolds numbers, the classical solutions constructed by Finn and Smith are unique in the class of D-solutions (that is, solutions with finite Dirichlet integral). No additional symmetry or decay assumptions are required

A Correction to this paper has been published: https://doi.org/10.1007/s00205-021-01608-9

In this paper we consider the regularity problem of the Navier–Stokes equations in \( {\mathbb {R}}^{3} \). We show that the Serrin-type condition imposed on one component of the velocity \( u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))\) with \( \frac{2}{p}+ \frac{3}{q} <1\), \( 3

We consider an analogue of the Lieb–Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb–Thirring constant converges to the optimal constant of the one-body Gagliardo–Nirenberg interpolation inequality without interaction.

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation

In this paper we prove the convergence of a suitable particle system towards the BGK model. More precisely, we consider an interacting stochastic particle system in which each particle can instantaneously thermalize locally. We show that, under a suitable scaling limit, propagation of chaos does hold and the one-particle distribution function converges to the solution of the BGK equation.

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form $$\begin{aligned} \mathcal {E}[V](X):=\sum _{1\leqq i\sqrt{2}\), in which case \({\overline{\mathcal E}_{\mathrm {sq}}[V]}=-4\). To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.

This paper considers the Alt–Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction \(e \in S^{d-1}\). In \(d=2\) we characterize the regularity properties