Flows with stagnation points, very challenging in analysis, are interesting and important phenomena in fluids. In this paper, we not only prove the uniqueness and existence of steady flows with a stagnation set, but also obtain the regularity of the boundary of the stagnation set, which is a class of obstacle-type free boundary. First, we prove a global uniqueness theorem for solutions of the two-dimensional steady Euler system, whose horizontal velocities tend to those of shear flows at the far field upstream. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non-Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the existence of steady incompressible Euler flows is established in an infinitely long nozzle via a variational approach. A very interesting phenomenon is that the boundary of a stagnation region can be regarded as an obstacle-type free boundary, which is proved to be globally C^{1} . Finally, the existence of a stagnation set is proved as long as the nozzle is wider than the width of the nozzle upstream, where the flows tend to Euler flows with stagnation points.
In this corrigendum, we correct an inaccuracy in the proofs of Propositions 2.5 and 2.6 in Ann. Inst. H. Poincaré C Anal. Non Linéaire 41, 1239–1287 (2024). Indeed, to obtain the refined error terms q_{i,\varepsilon} and p_{i,\varepsilon} we subtract only a radial term in Ann. Inst. H. Poincaré C Anal. Non Linéaire 41, 1239–1287 (2024), but in fact to ensure the desired properties of q_{i,\varepsilon} and p_{i,\varepsilon} one needs to subtract the whole Hessian. This change does not affect the main result of Ann. Inst. H. Poincaré C Anal. Non Linéaire 41, 1239–1287 (2024), Theorem 1.1, up to the expression of the refined local asymptotics which must include all second derivatives. Moreover, the mentioned refinement can be accommodated in a natural way into the existing proofs of Proposition 2.5 and 2.6 given in Ann. Inst. H. Poincaré C Anal. Non Linéaire 41, 1239–1287 (2024).
This article concerns the mathematical justification of an averaged system of partial differential equations governing the evolution of a two-phase mixture of compressible ideal fluids, with viscosity and without conductivity, in space dimension 1 with periodic boundary conditions. The derivation is done by some homogenization procedure. The originality of the paper consists in the fact that both the density and temperature are allowed to oscillate (because of the absence of heat conduction), so that the limiting model is a six-equation, two-pressure, two-temperature model. The key point is to show the strong convergence of the stress tensor in L-2((0,T) x (0, 1)). The main difficulties are obtaining uniform estimates in spite of the presence of oscillating coefficients in the energy equation. It requires looking at solutions with low regularity for the density and the temperature.
For any p is an element of (1, +infinity), we give a new inequality for the first nontrivial Neumann eigenvalue mu(p)(Omega, Phi) of the p-Laplacian on a convex domain Omega subset of R-N with a power-concave weight Phi. Our result improves the classical estimate in terms of the diameter, first stated in a seminal paper by Payne and Weinberger: we add into the lower bound an extra term depending on the second largest John semiaxis of Omega (equivalent to a power of the width in the special case N = 2). The power exponent in the extra term is sharp, and the constant in front of it is explicitly tracked, thus enlightening the interplay between space dimension, nonlinearity, and power concavity. Moreover, we attack the stability question: we prove that, if mu(p)(Omega, Phi) is close to the lower bound, then asymptotically Omega is close to a thin cylinder, and Phi is close to a function which is constant along its axis. As intermediate results, we establish a sharp L-infinity estimate for the associated eigenfunctions, and we determine the asymptotic behavior of mu(p)(Omega, Phi) for varying weights and domains, including the case of collapsing geometries.
In this paper we study the asymptotic stability threshold of the Couette flow for the 2D Navier-Stokes equations in a finite channel Omega=T & times;[-1,1] with Navier-slip boundary condition. It was proved that if the initial velocity v(0) satisfies parallel to v(0)-(y,0)parallel to(4)(H)<= c nu(1/3) for some small c independent of the viscosity coefficient nu, then the solution of the 2D Navier-Stokes equations rapidly converges to some shear flow close to Couette flow for t >>nu(-1/3.) Moreover, we prove the optimal enhanced dissipation and inviscid damping estimates. To this end, we develop a new approach that does not rely on the construction of the Fourier multiplier. Therefore, our approach opens a way toward the asymptotic stability threshold problem for other laminar flows in a domain with a physical boundary.
With the help of a new Picone-type identity, we prove that positive solutions u is an element of H-s(R-N) to the equation (-Delta)(s)u + u = u(p) in R(N )are nonradially nondegenerate, for all s is an element of (0, 1), N > 1 and p > 1 strictly smaller than the critical Sobolev exponent. By this we mean that the linearized equation (-Delta)(s)w + w-p(up-1)w = 0 does not admit nonradial solutions besides the directional derivatives of u. Letting B be the unit centered ball and )lambda(1)(B) the first Dirichlet eigenvalue of the fractional Laplacian (-Delta)(s), we also prove that positive solutions to (-Delta)(s)u + )lambda u = u(p) in B, with u = 0 on R-N \ B, are nonradially nondegenerate for any lambda >- lambda(1)(B) in the sense that the linearized equation does not admit nonradial solutions. From these results, we then deduce uniqueness and full nondegeneracy of positive solutions in some special cases. In particular, in the case N = 1, we prove that the equation (-Delta)(s)u + u = u(2) in R or in B, with zero exterior data, admits a unique even solution which is fully nondegenerate in the optimal range s E (16, 1), thus extending the classical uniqueness result of Amick and Toland on the Benjamin-Ono equation. Moreover, in the case N = 1, ) = 0, we also prove the uniqueness and full nondegeneracy of positive solutions for the Dirichlet problem in B with arbitrary subcritical exponent p. Finally, we determine the unique positive ground state solution of (-Delta)(1/)2u + u = u(p) in R-N, N > 1 with p = 1 + 2/ (N+1) and compute the sharp constant in the associated Gagliardo-Nirenberg inequality ||u||Lp+1.R-N/ <= CSI(-Delta) (1/4)u |(|N/N+2) L-2.R-N /.( 2)/N+2L(2)(R-N)
We consider the compressible Euler equation with a Coriolis term and prove a lower bound on the time of existence of solutions in terms of the speed of rotation, sound speed, and size of the initial data. Along the way, we obtain precise dispersive decay estimates for the linearized equation. In the incompressible limit, this improves current bounds for the incompressible Euler-Coriolis system as well.
We study the vortex formation in extreme type-II superconductors immersed in strong magnetic fields in the framework of the Ginzburg-Landau theory. We focus on the regime where superconductivity survives in the bulk of the material but the magnetic field penetrates the sample, i.e., for an applied field much larger than the first critical one, but below the transition to surface superconductivity. Through a two-scale vortex construction, we obtain precise estimates for the vortex distribution and prove the existence of isolated defects with non-trivial winding numbers. In this respect, our work provides the first rigorous mathematical proof of the existence of isolated vortices for fields comparable to the second critical one.
We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the standard CFL condition h similar to epsilon(2), where h is the time discretization step and epsilon the space step. We discuss extensions to more general convolution/redistancing schemes.
We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large amplitude. In the latter case, solutions may actually fail to exist for large amplitude, but in this case we prove that one of three physically meaningful breakdown scenarios occurs. Through the use of implicit function theorem techniques and a priori estimates, we construct both spatially periodic and solitary (non-periodic but spatially localized) solutions. The solitary case is substantially more complicated, requiring a delicate analysis in weighted Sobolev spaces. To the best of our knowledge, these results constitute the first general construction of stationary wave solutions, large or otherwise, to the viscous shallow water equations and the first general analysis of large solitary wave solutions to any viscous free boundary fluid model.
This paper is concerned with the asymptotic analysis of a sequence of variational models of brittle damage in the context of linearized elasticity in the two-dimensional discrete setting. We consider a discrete version of Francfort and Marigo's brittle damage model, where the total energy is restricted to continuous and piecewise affine vectorial displacements, within different regimes where the damaged regions concentrate on vanishingly small sets while the stiffness of the damaged material degenerates to 0. In this setting, the convergence of the space discretization, the concentration of the damaged regions, and the decay of the elastic properties of the damaged phase all compete simultaneously in non-trivial ways according to the scaling law under consideration. The mesh size turns out to be a crucial feature of the analysis, as it induces a minimal scale of spatial oscillations for admissible displacements. This study was motivated by the numerical investigations performed in Allaire-Jouve-Van Goethem (2011) on the one hand, where the authors have shown that forcing the stiffness decay on sets of arbitrarily small measure seems to lead to concentration phenomena such as in brittle fracture, and by the static analysis performed in Babadjian-Iurlano-Rindler (2021) on the other hand, where the authors addressed the rigorous asymptotic analysis of this observation in terms of the Gamma-convergence of the total energies, in the continuous setting in space. Surprisingly, they showed that fracture-type models were not obtained asymptotically, thus raising the question of the dependence of the effective models with respect to the scaling of a spatial discretization in Francfort and Marigo's model. This is the content of the present work, where we show that concentration phenomena are only captured asymptotically in regimes where the mesh size and the concentration of damaged regions are of the same order.
This work is concerned with the generation of decay estimates in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff on a bounded spatial domain for hard and moderately soft potentials. We work with a suitable notion of weak solutions, provided that mass, energy and entropy density functions are under control. We treat several boundary conditions: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. We show that the solutions generate some amount (up to d+1 ) of pointwise polynomial velocity decay. In the case of moderately soft potentials, we show that it is not possible to generate a decay higher than d+2 if the energy is bounded.
In this paper, we study the local well-posedness of nonlinear Schrödinger equations on tori \mathbb{T}^{d} at the critical regularity. We focus on cases where the nonlinearity |u|^{a}u is nonalgebraic with small a>0 . We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering a , we prove a bilinear estimate for the Schrödinger operator on tori \mathbb{T}^{d} , which enhances previously known multilinear estimates. We design a function space adapted to the new bilinear estimate and a package of Strichartz estimates, which is not based on conventional atomic spaces.
The Vlasov-Poisson system for ions is a kinetic equation for dilute, unmagnetised plasma. It describes the evolution of the ions in a plasma under the assumption that the electrons are thermalised. Consequently, the Poisson coupling for the electrostatic potential contains an additional exponential non-linearity not present in the electron Vlasov-Poisson system. The system can be formally derived through a mean-field limit from a microscopic system of ions interacting with a thermalised electron distribution. However, it is an open problem to justify this limit rigorously for ions modelled as point charges. Existing results on the derivation of the three-dimensional ionic Vlasov-Poisson system, obtained by the author and Iacobelli [J. Math. Pures Appl. 135 (2020), 199-255], require a truncation of the singularity in the Coulomb interaction at spatial scales of order N-beta with beta < 1/15, which is more restrictive than the available results for the electron Vlasov-Poisson system. In this article, we prove that the Vlasov-Poisson system for ions can be derived from a microscopic system of ions and thermalised electrons with interaction truncated at scale N-beta with beta < 1/3. We develop a generalisation of the probabilistic approach to mean-field limits developed in the works of Boers and Pickl [J. Stat. Phys. 164 (2016), 1-16] and Lazarovici and Pickl [Arch. Ration. Mech. Anal. 225 (2017), 1201-1231] that is applicable to interaction forces defined through a non-linear coupling with the particle density. The proof is based on a quantitative uniform law of large numbers for convolutions between empirical measures of independent, identically distributed random variables and locally Lipschitz functions.
We prove stochastic homogenization for a class of non-convex and non-coercive first-order Hamilton-Jacobi equations in a finite-range-dependence environment for Hamiltonians that can be expressed by a max-min formula. Exploiting the representation of solutions as value functions of differential games, we develop a game-theoretic approach to homogenization. We furthermore extend this result to a class of Lipschitz Hamiltonians that need not admit a global max-min representation. Our methods allow us to get a quantitative convergence rate for solutions with linear initial data toward the corresponding ones of the effective limit problem.
For s∈ (0,1) we introduce a notion of fractional s-mass on (n-2)-dimensional closed, orientable surfaces in ^n. Moreover, we prove its Γ-convergence, with respect to the flat topology, and pointwise convergence to the (n-2)-dimensional area.
We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$\partial_t u+\mathrm{div}\big(f(u)\big)=\mathcal{L}[b(u)]$$on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion operator $\mathcal{L}$ can be any symmetric L{\'e}vy operator (e.g. fractional Laplacians) and $b$ is nondecreasing and allowed to have degenerate regions ($b'=0$). We propose an entropy solution formulation for the problem and show uniqueness and existence of bounded entropy solutions under general assumptions. The uniqueness proof is based on the Kru\v{z}kov doubling of variables technique and incorporates several a priori results derived from our entropy formulation: an $L^\infty$-bound, an energy estimate, strong initial trace, weak boundary traces, and a \textit{nonlocal} boundary condition. The existence proof is based on fixed point iteration for zero-order operators $\mathcal{L}$, and then extended to more general operators through approximations, weak-$\star$ compactness of approximate solutions $u_n$, and \textit{strong} compactness of $b(u_n)$. Strong compactness follows from energy estimates and arguments we introduce to transfer weak regularity from $\partial_t u_n$ to $\partial_t b(u_n)$.Our work can be seen as both extending nonlocal theories from the whole space to domains and local theories on domains to the nonlocal case. Unlike local theories our formulation does not assume energy estimates. They are now a consequence of the formulation, and as opposed to previous nonlocal theories, play an essential role in our arguments. Several results of independent interest are established, including a characterization of the $\mathcal{L}$'s for which the corresponding energy/Sobolev-space compactly embeds into $L^2$.
We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential K. We are interested in establishing their well-posedness theory when the nonlocal interaction potential K is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that del K & lowast;K is in L-2, we can prove that the strong solution is unique. When K is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the n-species system.
In this paper we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in Rd. Our focus is on a broader category of densities, specifically those that are d1-concave and can be represented as V-d, where V is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem.
This paper deals with quasi-local isoperimetric versions of the positive mass theorem on 3-manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in such low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying C^0-stable quantitative estimates.
