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The thermodynamic uncertainty relation offers a universal energetic constraint on the relative magnitude of current fluctuations in nonequilibrium steady states. However, it has only been derived for long observation times. Here, we prove a…

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of…

The classical heat equation is incompatible with relativity, since the strong maximum principle allows for disturbances to propagate instantaneously. Some authors have proposed limiting the propagation speed by adding a linear hyperbolic…

The first-order general relativistic theory of a generic dissipative (heat-conducting, viscous, particle-creating) fluid is rediscussed from a unified covariant frame-independent point of view. By generalizing some previous works in the…

We are concerned with the time decay rates of strong solutions to a non-conservative compressible viscous two-phase fluid model in the whole space R3. Compared to the previous related works, the main novelty of this paper lies in the fact…

We consider Schr\"{o}dinger equations with real quadratic Hamiltonians, for which the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition.…

We extend a class of recently derived thermodynamic uncertainty relations to vector-valued observables. In contrast to the scalar-valued observables examined previously, this multidimensional thermodynamic uncertainty relation provides a…

The relativistic continuity equations for the extensive thermodynamic quantities are derived based on the divergence theorem in Minkowski space outlined by St\"uckelberg. This covariant approach leads to a relativistic formulation of the…

In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs.…

In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint…

Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute…

Lagrange and Hamilton equations for thermodynamic evolution near equilibrium as well as Schrodinger-like equation for the non-equilibrium case are obtained extending the CPDQ Principle (Constancy of the product momentum-oordinate…

We discuss the Generalized Uncertainty Principle and the Extended Uncertainty Principle in the context of black hole solutions coming from non-local theories of gravity, focusing, specifically, on Infinite Derivative Gravity. We argue that…

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation $\partial_tu-\Delta u=|u|^\alpha u,\;\alpha>0$ with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular,…

In the standard form of the relativistic heat equation used in astrophysics, information propagates instantaneously, rather than being limited by the speed of light as demanded by relativity. We show how this equation nonetheless follows…

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality \[ |\Delta u - u_t| \leq \frac{M}{|x|^2} |u|, \] with the critical inverse square potential. Our main…

Under Wigdersons' framework and by sorting out the technical points in the recent works of Tang (J. Fourier Anal. Appl. 31 (2025)) and Dias-Luef-Prata (J. Math. Pures Appl. (9) 198 (2025)), we prove an abstract uncertainty principle for…

We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of…

We develop a method for the transfer of an uncertainty principle for the short-time Fourier transform or a Fourier pair to an uncertainty principle for a sesquilinear or quadratic metaplectic time-frequency representation. In particular, we…

We study the problem of heat conduction in general relativity by using Carter's variational formulation. We write the creation rates of the entropy and the particle as combinations of the vorticities of temperature and chemical potential.…