Nonlinear Analysis-Theory Methods & Applications最新文献

We will present some rigidity results for solutions to semilinear elliptic equations of the form $\Delta u=W^{\prime }\left(u\right)$, where $W$ is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouville-type theorems and symmetry results, which generalise some known facts about the Cahn–Hilliard equation.

We prove the existence of a class of large global scattering solutions of Boltzmann’s equation with constant collision kernel in two dimensions. These solutions are found for $L^2$ perturbations of an underlying initial data which is Gaussian jointly in space and velocity. Additionally, the perturbation is required to satisfy natural physical constraints for the total mass and second moments, corresponding to conserved or controlled quantities. The space $L^2$ is a scaling critical space for the equation under consideration. If the initial data is Schwartz then the solution is unique and again Schwartz on any bounded time interval.

In this paper, we study the global regularity problem for the 2D Rayleigh–Bénard equations with logarithmic supercritical dissipation. By exploiting a combined quantity of the system, the technique of Littlewood-Paley decomposition and Besov spaces, and some commutator estimates, we establish the global regularity of a strong solution to this equations in the Sobolev space $H^s\left(R^2\right)$ for $s\ge 2$.

We consider codimension 1 area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\Sigma$ of Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt}\left(T\right)\setminus {\mathrm{spt}}^p\left(\partial T\right)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in Minter and Wickramasekera (2024) as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $\Sigma$. This improvement is in fact crucial in De Lellis et al., (2022) to prove that the singular set of $T$ can be decomposed into a $C^{1,\alpha }$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.

The Cauchy problem for hyperbolic systems of balance laws admits global smooth solutions near the constant states under stability condition. This was widely studied in previous works. In this paper, we concern hyperbolic systems with time-dependent damping $\mu {(1+t)}^{-\lambda }G\left(U\right)$ with $\mu >0,\lambda >0$. In the following two cases, $\left(i\right)\lambda =1,\mu >{\mu }_0,$ where ${\mu }_0>0$ is a constant depending only on the coefficients of the system; $\left(ii\right)0<\lambda <1,\mu >0,$ we prove that the smooth solutions exist globally when the initial data is small. To obtain these stability results, we establish uniform energy estimates and various dissipative estimates for all time and employ an induction argument on the order of derivatives of smooth solutions. Finally, we apply these results to some physical models.

In this article, we extend the foundations of the theory of differential inclusions in the space of compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context, we prove a novel existence result à la Peano for this class of dynamics under mere Carathéodory regularity assumptions. The latter is based on a natural, yet previously unexplored set-valued adaptation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations whose right-hand sides are measurable in the time variable. By leveraging some of the underlying methods along with new estimates for solutions of continuity equations, we also bring substantial improvements to the earlier versions of the Filippov estimates, compactness and relaxation properties of the solution sets of continuity inclusions, which are derived in the Cauchy–Lipschitz framework.

In this paper, we investigate the existence, uniqueness and stability of forced waves for the Fisher–KPP equation in a shifting environment without imposing the monotonicity condition on the shifting intrinsic growth term. First, the existence of forced waves for some range of shifting speeds is proved. Then we prove the uniqueness of saturation forced waves. Moreover, a new method is introduced to derive the non-existence of forced waves. Finally, we derive the stability of forced waves under certain perturbation of a class of initial data.

We study the large time asymptotics of solutions to the Cauchy problem for the modified Korteweg–de Vries-Benjamin–Ono equation $\left(\begin{array}{c}{\partial }_{t}u+\frac{a}{2}H{\partial }_x^{2}u-\frac{b}{3}{\partial }_x^{3}u={\partial }_x\left(u^3\right),t>0,x\in R,u\left(0,x\right)=u_0\left(x\right),x\in R,\end{array}\right)$where $a,b>0,$ $H\varphi =\frac{1}{\pi }$p.v.${\int }_R\frac{\varphi \left(y\right)}{x-y}dy$ is the Hilbert transform. We develop the factorization technique to obtain the sharp time decay estimate for solutions and to prove the modified scattering.

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\Omega \subset R^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano (Conti et al., 2017; Conti et al., 2019) for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.

In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the classical fractional Laplacian as the mean value (in the sphere) of the 1-dimensional fractional Laplacians in lines with directions in the sphere. To obtain this second new fractional operator we just replace the mean value by the mid-range of 1-dimensional fractional Laplacians with directions in the sphere. For these two new fractional operators we prove a comparison principle for viscosity sub and supersolutions and then we obtain existence and uniqueness for the Dirichlet problem, that turns out to be nonlinear. Strong maximum and comparison principles also hold. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as $s\nearrow 1$, while for the second operator we obtain the sum of the smallest and the largest classical Hessian eigenvalues.