# Analysis of PDEs

## New submissions

[ total of 28 entries: 1-28 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case
Subjects: Analysis of PDEs (math.AP)

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in \cite{MartelM:Instab:gKdV} to show the orbital instability of the solitary waves $\Q\sts{x-ct}\e^{\i\omega t}$ of the generalized derivative nonlinear Schr\"odinger equation (gDNLS) in the degenerate case $c=2z_0\sqrt{\omega}$, where $z_0=z_0\sts{\sigma}$ is the unique zero point of $F\sts{z;~\sigma}$ in $\sts{-1, ~ 1}$. The new ingredients in the proof are the refined modulation decomposition of the solution near $\Q$ according to the spectrum property of the linearized operator $\Scal_{\omega, c}"\sts{\Q}$ and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.

[2]
Title: Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems
Subjects: Analysis of PDEs (math.AP)

We study the local continuity and asymptotic behavior of solutions, $\bold{u}=(u^1,\cdots, u^k)$, of degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$} \end{equation*} describing the degenerate diffusion of the populations density vector, $\bold{u}$, of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. We adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto for the local continuity of solutions, $u^i$. Under some regularity condition, we also prove that the population density function, $\bold{u}$, of $i$-th species with the population $M_i$ converges to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ in the space of differentiable functions of all order where $\mathcal{B}_M$ is the Barenblatt profile of the Porous Medium Equation with $L^1$ mass $M=M_1+\cdots+M_k$ while $U$ converges to $\mathcal{B}_M$. As a consequence, each $u^i$ becomes a concave function after a finite time.

[3]
Title: The nodal set of solutions to some elliptic problems: singular nonlinearities
Subjects: Analysis of PDEs (math.AP)

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. We extend to this class of \emph{singular} equations the results recently obtained in \cite{SoTe2018} for \emph{sublinear and discontinuous} equations, $1\leq q<2$, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most $N-2$ (locally finite when $N=2$). As an intermediate step, we establish the regularity of a class of \emph{not necessarily minimal} solutions.
The proofs are based on a priori bounds, monotonicity formul\ae \ for a $2$-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.

[4]
Title: Local-in-time well-posedness for Compressible MHD boundary layer
Subjects: Analysis of PDEs (math.AP)

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work \cite{liu2016mhdboundarylayer} on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.

[5]
Title: Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling
Comments: Submitted version in Discrete Contin. Dyn. Syst. Ser. B
Subjects: Analysis of PDEs (math.AP)

Hysteresis is an important issue in modeling piezoelectric materials, for example, in applications to energy harvesting, where hysteresis losses may influence the efficiency of the process. The main problem in numerical simulations is the inversion of the underlying hysteresis operator. Moreover, hysteresis dissipation is accompanied with heat production, which in turn increases the temperature of the device and may change its physical characteristics. More accurate models therefore have to take the temperature dependence into account for a correct energy balance. We prove here that the classical Preisach operator with a fairly general parameter-dependence admits a Lipschitz continuous inverse in the space of right-continuous regulated functions, propose a time-discrete and memory-discrete inversion algorithm, and show that higher regularity of the inputs leads to a higher regularity of the output of the inverse.

[6]
Title: High frequency limits for invariant Ruelle densities
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Spectral Theory (math.SP)

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.

[7]
Title: Determining both the source of a wave and its speed in a medium from boundary measurements
Subjects: Analysis of PDEs (math.AP)

We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if $c^{-2}$ is harmonic in $\omega \subset R^3$ and identically 1 on $\omega^c$, where $\omega$ is a simply connected region, then a non-trapping wave speed $c$ can be uniquely determined from the solution of the wave equation on boundary of $\Omega \supset \supset \omega$ without the knowledge of the source. We also show that if the wave speed $c$ is known and only assumed to be bounded then, under mild assumptions on the set of discontinuous points of $c$, the source of the wave can be uniquely determined from boundary measurements.

[8]
Title: On stabilization of solutions of higher order evolution inequalities
Subjects: Analysis of PDEs (math.AP)

We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $$\sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) - u_t \ge f (x, t) g (u) \quad \mbox{in} {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), \quad m,n \ge 1,$$ stabilizes to zero as $t \to \infty$. These conditions generalize the well-known Keller-Osserman condition on the grows of the function $g$ at infinity.

[9]
Title: Boosting the Maxwell double layer potential using a right spin factor
Authors: Andreas Rosén
Subjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)

We construct new spin singular integral equations for solving scattering problems for Maxwell's equations, both against perfect conductors and in media with piecewise constant permittivity, permeability and conductivity, improving and extending earlier formulations by the author. These differ in a fundamental way from classical integral equations, which use double layer potential operators, and have the advantage of having a better condition number, in particular in Fredholm sense and on Lipschitz regular interfaces, and do not suffer from spurious resonances. The construction of the integral equations builds on the observation that the double layer potential factorises into a boundary value problem and an ansatz. We modify the ansatz, inspired by a non-selfadjoint local elliptic boundary condition for Dirac equations.

[10]
Title: Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition
Journal-ref: Topol. Methods Nonlinear Anal. 51:1 (2018), 55-77
Subjects: Analysis of PDEs (math.AP)

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: $-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }\Omega,$ and $u=0,\text{ on }\partial \Omega.$ We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. Our results extend previous work by Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comp. Math. Appl. 69 (2015), 1-12, and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

[11]
Title: Note on Calderón's inverse problem for measurable conductivities
Subjects: Analysis of PDEs (math.AP)

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (\sigma \nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calder\'on problem in three and higher dimensions in the $L^\infty$ case.

[12]
Title: Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
Authors: Jonas M. Tölle
Subjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA); Probability (math.PR)

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with $C^{1}$-vector field coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^o-formula.

[13]
Title: Asymptotic behaviour of neuron population models structured by elapsed-time
Subjects: Analysis of PDEs (math.AP)

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In the first model, the structuring variable $s$ represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic "state". We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e., weak connectivity). The main innovation is the use of Doeblin's theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.

### Cross-lists for Tue, 20 Mar 18

[14]  arXiv:1803.06432 (cross-list from math.FA) [pdf, ps, other]
Title: Pseudo-differential operators with nonlinear quantizing functions
Subjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Operator Algebras (math.OA)

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi,$$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In particular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and $\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $\tau$ and here we investigate the corresponding calculus in the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.

[15]  arXiv:1803.06543 (cross-list from math.PR) [pdf, ps, other]
Title: The parametrix method for parabolic SPDEs
Comments: Submitted to Stochastics and Partial Differential Equations: Analysis and Computations
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in H\"older classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the It\^o-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

[16]  arXiv:1803.06697 (cross-list from math.DG) [pdf, ps, other]
Title: Higher-order estimates for collapsing Calabi-Yau metrics
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Complex Variables (math.CV)

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform C^infinity estimate. We then apply these local results to the case of collapsing Calabi-Yau metrics on compact Calabi-Yau manifolds. In this global setting, the C^0 estimate required as a hypothesis in our new local C^alpha and C^infinity estimates is known to hold thanks to earlier work of the second-named author.

[17]  arXiv:1803.06953 (cross-list from math.PR) [pdf, ps, other]
Title: Entropy solutions for stochastic porous media equations
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\Delta (|u|^{m-1}u)$ for all $m\in(1,\infty)$, and H\"older continuous diffusion nonlinearity with exponent $1/2$.

### Replacements for Tue, 20 Mar 18

[18]  arXiv:1609.07232 (replaced) [pdf, ps, other]
Title: From visco to perfect plasticity in thermoviscoelastic materials
Authors: Riccarda Rossi
Comments: To appear in Zeitschrift fur Angewandte Mathematik und Mechanik
Subjects: Analysis of PDEs (math.AP)
[19]  arXiv:1704.07174 (replaced) [pdf, ps, other]
Title: On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$
Authors: Robert Schippa
Comments: 33 pages, existence of solutions and a priori estimates for large initial data added
Subjects: Analysis of PDEs (math.AP)
[20]  arXiv:1705.02457 (replaced) [pdf, ps, other]
Title: On nonlinear cross-diffusion systems: an optimal transport approach
Comments: improved version; some well-known results shortened
Subjects: Analysis of PDEs (math.AP)
[21]  arXiv:1708.06172 (replaced) [pdf, ps, other]
Title: Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism
Authors: Yi Zhu
Subjects: Analysis of PDEs (math.AP)
[22]  arXiv:1712.07768 (replaced) [pdf, other]
Title: Electric field concentration in the presence of an inclusion with eccentric core-shell geometry
Subjects: Analysis of PDEs (math.AP)
[23]  arXiv:1802.00215 (replaced) [pdf, other]
Title: Discontinuous traveling waves as weak solutions to the Fornberg-Whitham equation
Subjects: Analysis of PDEs (math.AP)
[24]  arXiv:1802.05502 (replaced) [pdf, ps, other]
Title: Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator
Subjects: Analysis of PDEs (math.AP)
[25]  arXiv:1803.02040 (replaced) [pdf, ps, other]
Title: Principal eigenvalues of a class of nonlinear integro-differential operators
Authors: Anup Biswas
Subjects: Analysis of PDEs (math.AP)
[26]  arXiv:1803.03231 (replaced) [pdf, other]
Title: Principal eigenvalue and maximum principle for cooperative periodic-parabolic systems
Comments: 38 pages, 2 figures, Remarks on the generation of the paper at the end of the Introduction
Subjects: Analysis of PDEs (math.AP)
[27]  arXiv:1803.06299 (replaced) [pdf, other]
Title: On the existence of a scalar pressure field in the Bredinger problem