Analysis of PDEs (math.AP)

Abstract: In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{\gamma/2} a^{i'n} u_{i'n} + x_n^{\gamma} a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{\gamma/2} b^n u_{n} + c u + f \quad (\gamma \leq1).$$ When the equation above is singular, it can be derived from Monge--Amp\`ere equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution $u$, which is called $s$-polynomial. To prove $C_s^{2+\alpha}$-regularity and higher regularity of the solution $u$, we establish generalized Schauder theory which approximates coefficients of the operator with $s$-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap method to obtain higher regularity.

Abstract: This paper is concerned with existence, non-existence and uniqueness of positive (coexistence) steady states to a predator-prey system with density-dependent dispersal. To overcome the analytical obstacle caused by the cross-diffusion structure embedded in the density-dependent dispersal, we use a variable transformation to convert the problem into an elliptic system without cross-diffusion structure. The transformed system and pre-transformed system are equivalent in terms of the existence or non-existence of positive solutions. Then we employ the index theory alongside the method of the principle eigenvalue to give a nearly complete classification for the existence and non-existence of positive solutions. Furthermore we show the uniqueness of positive solutions and characterize the asymptotic profile of solutions for small or large diffusion rates of species. Our results pinpoint the positive role of density-dependent dispersal on the population dynamics for the first time by showing that the density-dependent dispersal is a beneficial strategy promoting the coexistence of species in the predator-prey system by increasing the chance of predator's survival.

Abstract: Motivated by an application involving additively manufactured bioresorbable polymer scaffolds supporting bone tissue regeneration, we investigate the impact of uncertain geometry perturbations on the effective mechanical properties of elastic rods. To be more precise, we consider elastic rods modeled as three-dimensional linearly elastic bodies occupying randomly perturbed domains. Our focus is on a model where the cross-section of the rod is shifted along the longitudinal axis with stationary increments. To efficiently obtain accurate estimates on the resulting uncertainty of the effective elastic moduli, we use a combination of analytical and numerical methods. Specifically, we rigorously derive a one-dimensional surrogate model by analyzing the slender-rod $\Gamma$-limit. Additionally, we establish qualitative and quantitative stochastic homogenization results for the one-dimensional surrogate model. To compare the fluctuations of the surrogate with the original three-dimensional model, we perform numerical simulations by means of finite element analysis and Monte Carlo methods.

Abstract: In this article, we investigate the incoming boundary value problem for the stationary linearized Boltzmann equations in $ \Omega \subseteq \mathbb{R}^{3}$. For a $C^2$ bounded domain with boundary of positive Gaussian curvature, the existence theory is established in $H^{1}(\Omega \times \mathbb{R}^{3})$ provided that the diameter of the domain $\Omega$ is small enough.

Abstract: In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn-Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fr\'echet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions.

Abstract: Dealing with the inverse source problem for the scalar wave equation, we have shown recently that we can reconstruct the spacetime dependent source function from the measurement of the wave, collected on a single point $x$ and a large enough interval of time, generated by a small scaled bubble, enjoying large contrasts of its bulk modulus, injected inside the domain to image. Here, we extend this result to reconstruct not only the source function but also the variable wave speed. Indeed, from the measured waves, we first localize the internal values of the travel time function by looking at the behavior of this collected wave in terms of time. Then from the Eikonal equation, we recover the wave speed. Second, we recover the internal values of the wave generated only by the background (in the absence of the small particles) from the same measured data by inverting a Volterra integral operator of the second kind. From this reconstructed wave, we recover the source function at the expense of a numerical differentiation.

Abstract: We establish a convergence theorem for Crandall-Lions viscosity solutions to path-dependent Hamilton-Jacobi-Bellman PDEs. Our proof is based on a novel convergence theorem for dynamic sublinear expectations and the stochastic representation of viscosity solutions as value functions.

Abstract: We obtain uniform in time $L^\infty$-bounds for the solutions to a class of thermo-diffusive systems. The nonlinearity is assumed to be at most sub-exponentially growing at infinity and have a linear behavior near zero.

Abstract: We study the local-in-time well-posedness for an interface that separates an anisotropic plasma from a vacuum. The plasma flow is governed by the ideal Chew-Goldberger-Low (CGL) equations, which are the simplest collisionless fluid model with anisotropic pressure. The vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The plasma and vacuum magnetic fields are tangential to the interface. This represents a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. By a suitable symmetrization of the linearized CGL equations we reduce the linearized free boundary problem to a problem analogous to that in isotropic magnetohydrodynamics (MHD). This enables us to prove the local existence and uniqueness of solutions to the nonlinear free boundary problem under the same non-collinearity condition for the plasma and vacuum magnetic fields on the initial interface required by Secchi and Trakhinin (Nonlinearity 27:105-169, 2014) in isotropic MHD.

Abstract: We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant problem: Given a compact Riemannian manifold with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet})$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions, and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet})$ can be ``corrected" into a complex $(\mathcal{A}_{\bullet})$ of pseudodifferential operators, where $\mathcal{A}_k - A_k$ is a zero-order correction within this class. The induced complex $(\mathcal{A}_{\bullet})$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

Abstract: In this paper we study a recently derived mathematical model for nonlinear propagation of waves in the atmosphere, for which we establish the local well-posedness in the setting of classical solutions. This is achieved by formulating the model as a quasilinear parabolic evolution problem in an appropriate functional analytic framework and by using abstract theory for such problems. Moreover, for $L_2$-initial data, we construct global weak solutions by employing a two-step approximation strategy based on a Galerkin scheme, where an equivalent formulation of the problem in terms of a new variable is used. Compared to the original model, the latter has the advantage that the $L_2$-norm is a Liapunov functional.

Abstract: In this paper we study the upper bound of the curvature estimate for nodal sets of harmonic functions in the plane. Using the complex methods, we prove that at any non-critical point $p$, the curvature of any nodal curve of a harmonic function $u$ is upper bounded by $$ \left|\kappa(u)(p)\right|\leq \frac{C}{r} $$ where $u$ has only one nodal curve in $B_r(p)$ across $p$. L. De Carli and Steve M. Hudson proved that the constant $C\leq 24$. In this paper, we prove that the sharp upper bound $C$ is 8, and we also prove that the equality holds if and only if $u$ is a harmonic function related to some Koebe function. On the other hand, we obtain the curvature estimate of nodal curves of harmonic functions at critical points. Thus we prove that, for harmonic functions, the curvature of every nodal curve at any point $p$ is upper bounded by the distance between $p$ and other nodal curves, and the distance from $p$ to the boundary of the domain.

Abstract: The paper presents a complete (to the best of the author's knowledge) overview on the existing literature concerning the NLS equation with point-concentrated nonlinearity. Precisely, it mainly covers the following topics: definition of the model, weak and strong local well-posedness, global well-posedness, classification and stability (orbital and asymptotic) of the standing waves, blow-up analysis and derivation from the standard NLS equation with shrinking potentials. Also some related problem is mentioned.