Analysis of PDEs (math.AP)

Abstract: We have established (a weak form of) ill-posedness for the KdV-Burgers equation on a real line in Fourier amalgam spaces $\widehat{w}_s^{p,q}$ with $s<-1$. The particular case $p=q=2$ recovers the result of L. Molinet and F. Ribaud [Int. Math. Res. Not., (2002), pp. 1979-2005]. The result is new even in Fourier Lebesgue space $\mathcal{F}L_s^q$ which corresponds to the case $p=q(\neq 2)$ and in modulation space $M_s^{2,q}$ which corresponds to the case $p=2,q\neq 2$.

Abstract: In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

Abstract: We consider the following $p$ order nonlinear half wave Schr{\"o}dinger equations$$\left(i \partial\_{t}+\partial\_{x }^2-\left|D\_{y}\right|\right) u=\pm|u|^{p-1} u$$on the plane $\mathbb{R}^2$ with $1<p\leq 2$. This equation is considered as a toy model motivated by the study of solutions to weakly dispersive equations. In particular, the global well-posedness of this equation is a difficult problem due to the anisotropic property of the equation, with one direction corresponding to the half-wave operator, which is not dispersive. In this paper, we prove the global well-posedness of this equation in $L\_x^2 H\_y^s(\mathbb{R}^2) \cap H\_x^1 L\_y^2(\mathbb{R}^2)$($\frac{1}{2}\leq s \leq 1$), which is the first global well-posedness result of nonlinear half wave Schr{\"o}dinger equations. With the global well-posedness in the energy space for the focusing equation and the study on the solitary wave in [1], we complete the proof of the stability of the set of ground states. Moreover, we consider the half wave Schr{\"o}dinger equations on $\mathbb{R}\_{x}\times\mathbb{T}\_{y}$, which can also be called the wave guide Schr{\"o}dinger equations on $\mathbb{R}\_{x}\times\mathbb{T}\_{y}$. Using a similar approach in the analysis of the Cauchy problem of half wave Schr{\"o}dinger equations on $\mathbb{R}^2$, we can also deduce the global well-posedness of $p$ ($1<p\leq2$) order wave guide Schr{\"o}dinger equations in $L\_x^2 H\_y^s(\mathbb{R}\times\mathbb{T}) \cap H\_x^1 L\_y^2(\mathbb{R}\times\mathbb{T})$ with $\frac{1}{2}\leq s \leq 1$. With the global well-posedness in the energy space for the focusing wave guide Schr{\"o}dinger equations and the study on the ground states in [2], we complete the proof of the orbital stability of the ground states with small frequencies.

Abstract: Well-posedness in $L_\infty$ of the nonlocal Gray-Scott model is studied for integrable kernels, along with the stability of the semi-trivial spatially homogeneous steady state. In addition, it is shown that the solutions to the nonlocal Gray-Scott system converge to those to the classical Gray-Scott system in the diffusive limit.

Abstract: In 1987, Michael Wiegner in his seminal paper [17] provided an important result regarding the energy decay of Leray solutions $\boldsymbol u(\cdot,t)$ to the incompressible Navier-Stokes in $\mathbb{R}^{n}$: if the associated Stokes flows had their $\hspace{-0.020cm}L^{2}\hspace{-0.050cm}$ norms bounded by $O(1 + t)^{-\;\!\alpha} $ for some $ 0 < \alpha \leq (n+2)/4 $, then the same would be true of $ \|\hspace{+0.020cm} \boldsymbol u(\cdot,t) \hspace{+0.020cm} \|_{L^{2}(\mathbb{R}^{n})} $. The converse also holds, as shown by Z.Skal\'ak [15] and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier-Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two-sided bounds of the form $(1+t)^{-\alpha}\lesssim \| \boldsymbol u(\cdot,t)\|_{L^2(\mathbb{R}^n)} \lesssim (1+t)^{-\alpha}$.

Abstract: Extending work of Yang-Zumbrun for the hydrodynamically stable case of Froude number F < 2, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin-film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams are all rigorously obtained by mathematical analysis and explicit calculation.

Abstract: We investigate the existence of generalized solutions to coercive competing system driven by the (p,q) -Laplacian with unbounded perturbation corresponding to the leading term in the differential operator and with convection depending on the gradient. Some abstract principle leading to the existence of generalized solutions is also derived basing on the Galerkin scheme.

Abstract: In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.

Abstract: We introduce an analytic method which stably reconstructs both components of a (sufficiently) smooth, real valued, vector field compactly supported in the plane from knowledge of its Doppler transform and its first moment Doppler transform. The method of proof is constructive. Numerical inversion results indicate robustness of the method.

Abstract: For $s \in [1/2, 1)$, let $u$ solve $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^{n} \times [-T, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb R^n \times [-T, 0])} < \infty$. We show that if for some $0< c< T$ and $\epsilon>0$ $$\frac{1}{c} \int_{[-c,0]} u^2(x, t) dt \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb R^n,$$ then $u \equiv 0$ in $\mathbb R^{n} \times [-T, 0]$.

Abstract: We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kaj\'ant\'o, Krist\'aly, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.

Abstract: We provide well-posedness theory of a nonlinear structured population model on an abstract metric space which is only assumed to be separable and complete. To this end, we leverage the structure of the space of nonnegative Radon measures under the dual bounded Lipschitz distance (flat metric) which can be seen as a generalization of Wasserstein distance to nonconservative problems. Motivated by applications, the formulation of models on fairly general metric spaces allows us to consider processes on infinite-dimensional state spaces or on graphs combining discrete and continuous structures.