Abstract We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $$\textrm{BD}(\Omega )$$ BD ( Ω ) have weak gradients in $$\textrm{L}_{\textrm{loc}}^{1}(\Omega ;\mathbb {R}^{n\times n})$$ L loc 1 ( Ω ; R n × n ) . This is achieved for the sharp ellipticity range that is presently known to yield $$\textrm{W}_{\textrm{loc}}^{1,1}$$ W loc 1 , 1 -regularity in the full gradient case on $$\textrm{BV}(\Omega ;\mathbb {R}^{n})$$ BV ( Ω ; R n ) . As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on $$\textrm{BD}(\Omega )$$ BD ( Ω ) .
Abstract In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $$|\nabla |^{\alpha }$$ | ∇ | α for any $$\alpha \in [0, \alpha _0)$$ α ∈ [ 0 , α 0 ) ( $$\alpha _0 = \frac{22-8\sqrt{7}}{9} > 0$$ α 0 = 22 - 8 7 9 > 0 ). We construct solutions in $$\mathbb {R}^3\times [0,T]$$ R 3 × [ 0 , T ] with a finite $$T>0$$ T > 0 and with an external forcing which is in $$L^1_t([0, T]) C_x^{1,\epsilon }\cap L^{\infty }_{t} L^{2}_{x}$$ L t 1 ( [ 0 , T ] ) C x 1 , ϵ ∩ L t ∞ L x 2 , such that for each time $$t \in [0, T)$$ t ∈ [ 0 , T ) , the velocity u is in the space $$C^\infty \cap L^2$$ C ∞ ∩ L 2 and such that as the time t approaches the blow-up moment T , the integral $$\int _0^t |\nabla u| \text {d}s$$ ∫ 0 t | ∇ u | d s tends to infinity. Since this result lays in a well-posedness class, the blow-up is generated by the dynamics of the equation and not by the force itself. This is the first blow-up result for hypodissipative Navier–Stokes in a well-posedness class.
Abstract We provide a structural analysis of the space of functions of bounded deviatoric deformation, $$\textrm{BD}_{\textrm{dev}}$$ BD dev , which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $$\textrm{BD}_{\textrm{dev}}$$ BD dev -maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation, homogenization, and integral representation problems, allowing for integrands with explicit dependence on u as well as $${{\mathcal {E}}}_d u$$ E d u . Our approach overcomes several difficulties as compared to the $$\textrm{BD}$$ BD case, in particular due to the lack of invariance of $${{\mathcal {E}}}_d$$ E d under orthogonalization of the polar directions.
This paper investigates the long–time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of N interacting point particles, where the time evolution of the joint distribution f_N(t) is governed by the Liouville equation. Our primary objective is to analyze the system’s behavior over extended time intervals, focusing on the stability, the potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime N ≫ 1 , exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases we prove that it is possible to derive uniform bounds in L^2 , both over time and with respect to the number of particles, on the marginals ( f_k,N) _1≦ k ≦ N , irrespective of the initial state’s chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels K ∈ W^-2/d+2, d+2 in dimension d≧ 2 , surpassing the traditional L^d regularity barriers. Finally, we address the highly singular case of K ∈ H^-1 under a threshold temperature regime, offering new insights into the behavior of such systems.
In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative root term is given by |del|(alpha) for any alpha is an element of [0, alpha(0)) (alpha(0) = 22-8 root 7 / 9 > 0). We construct 7 solutions in R-3 & times; [0, T] with a finite T > 0 and with an external forcing which is in L-t(1) ([0, T])C-x(1,epsilon )boolean AND (LtLx2)-L-infinity, such that for each time t is an element of [0, T), the velocity u is in the space C-infinity boolean AND L-2 and such that as the time t approaches the blow-up moment T, the integral f(0)(t)|del u|ds tends to infinity. Since this result lays in a well-posedness class, the blow-up is generated by the dynamics of the equation and not by the force itself. This is the first blow-up result for hypodissipative Navier-Stokes in a well-posedness class.
We investigate some of the properties of the mapping from wave-functions to single particle densities, partially answering an open question posed by E. H. Lieb in 1983.
We investigate the role of the four viscosity parameters in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in L^2 , in the case of a spin viscosity γ≥ 0 , and we construct the asymptotic profiles of the solution as t→ +∞ . We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity μ , and not on the other parameters χ (vortex-viscosity), γ (spin viscosity) and κ (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times.
We establish that, in general, there are no entropy solutions to the Cauchy problem for the multidimensional Burgers equation when the initial data is a measure. This answers in the negative a conjecture of D. Serre–L. Silvestre (Arch Rat Mech Anal 234:1391–1411, 2019). This conjecture was motivated by the description of the asymptotic behavior of entropy solutions with integrable initial data. Despite this negative result, we still derive some information on the asymptotic behavior of such solutions. Our method relies on new Laplace transform estimates for the propagation of information.
In this paper we study how to determine if a linear biochemical network satisfies the detailed balance condition, without knowing the details of all the reactions taking place in the network. To this end, we use the formalism of response functions R_ij (t) that measure how the system reacts to the injection of the substance j at time t=0 , by measuring the concentration of the substance i j for t >0 . In particular, we obtain a condition involving two reciprocal measurements (that is R_ij(t) , R_ji(t) ) that is necessary, but not sufficient, for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied by the graph associated to the network, as well as a stability property that guarantees that the chemical rates are not fine-tuned.
We correct the statement of part (2) of Proposition 4.3, correct typographical errors, and clarify a point concerning equation (B.6) for ρ _g . Moreover, we provide an improved argument for the proof of the main result: Theorem 6.7. The idea of the proof is to reduce the quasilinear problem to a semilinear problem via the implicit function theorem.
In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri–De Giorgi–Miranda (Arch Rational Mech Anal 32:255–267, 1969) and Simon (Indiana Univ Math J 25:821–855, 1976) to an appropriate free boundary setting.
In this work, starting from the predictions of the Post-Newtonian theory for a system of N infalling masses from the infinite past i^- , we formulate and solve a scattering problem for the system of linearised gravity around Schwarzschild in a double null gauge, as introduced in Dafermos (Acta Math 222:1–214, 2019). The scattering data are posed on a null hypersurface 𝒞 emanating from a section of past null infinity ℐ^- , and on the part of ℐ^- that lies to the future for this section. Along 𝒞 , we implement the Post-Newtonian theory-inspired hypothesis that the gauge-invariant components of the Weyl tensor and (a.k.a. Ψ _0 and Ψ _4 ) decay like r^-3 , r^-4 , respectively, and we exclude incoming radiation from ℐ^- by demanding the News function to vanish along ℐ^- . We also show that compactly supported gravitational perturbations along ℐ^- induce very similar data, with , decaying like r^-3 , r^-5 . After constructing the unique solution to this scattering problem, we then provide a complete analysis of the asymptotic behaviour of projections onto fixed spherical harmonic number ℓ near ℐ^- , spacelike infinity i^0 and future null infinity ℐ^+ , crucially exploiting a set of approximate conservation laws enjoyed by and . Having obtained a clear understanding of the asymptotics of linearised gravity around Schwarzschild, we also give constructive corrections to popular historical notions of asymptotic flatness such as Bondi coordinates or asymptotic simplicity. In particular, confirming earlier heuristics authorized by Damour and Christodoulou, we find that the peeling property is violated both near ℐ^- and near ℐ^+ , with for example near ℐ^+ only decaying like r^-4 instead of r^-5 . We also find that the resulting solution decays slower towards i^0 than often assumed, with both decaying like r^-3 towards i^0 . The issue of summing up the estimates obtained for fixed angular modes in ℓ in order to obtain asymptotics for the full solution is dealt with in forthcoming work.
For the physical vacuum free boundary problem of the damped compressible Euler equations in both 2D and 3D, we prove the global existence of smooth solutions and justify their time-asymptotic equivalence to the corresponding Barenblatt self-similar solutions derived from the porous media equation under a Darcy’s law approximation, provided that the initial data are small perturbations of the Barenblatt solutions. Building on the 3D almost global existence result in [Zeng, Arch. Ration. Mech. Anal. 239, 553–597 (2021)], our key contribution lies in improving the decay rate of the time derivative of the perturbation from -1 (as previously established) to -1-ε for a fixed constant ε > 0 . This critical enhancement ensures time integrability and hence global existence. Together with the previous 1D result in [Luo–Zeng, Comm. Pure Appl. Math. 69, 1354–1396 (2016)], the results obtained in this paper provide a complete answer to the question raised in [Liu, T.-P.: Jpn. J. Appl. Math. 13, 25–32 (1996)]. Moreover, we also consider the problem with time-dependent damping of the form (1+t)^-λ for 0< λ < 1 . Notably, our framework unifies the treatment of both time-dependent ( 0< λ < 1 ) and time-independent ( λ = 0 ) damping cases across dimensions. We further quantify the decay rates of the density and velocity, as well as the expansion rate of the physical vacuum boundary.
We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the p-Laplace equation -ε ^p Δ _p u = u^q-1 - u^p-1 in Ω , where Ω is a bounded domain in ℝ^n with C^2 -boundary, 1
For any initial datum θ _0∈ L^4/3_x it is proven that the existence of a global-in-time weak solution θ∈ L^∞ _t L^4/3_x to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the Ḣ^-1/2_x norm, is constant in time. The solution is obtained as a vanishing viscosity limit. The main idea is to propagate in time the non-concentration of the L^4/3_x norm of the initial data, from which the strong compactness in the Hamiltonian norm is deduced. Minimal Onsager supercritical conditions preventing anomalous dissipation are given.
We prove that certain renormalized value functions associated with the d-dimensional ( d≧ 2 ) N-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton–Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its ℋ^d(N-1)-1 -rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.
We study scattering for the linear Helmholtz operator in two dimensions and develop a technique, which can be used to ascertain scattering of a given incident wave from very regular inhomogeneities. This technique is then applied to a number of interesting examples.
