13 research outputs found
Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the Prodi-Serrin condition
No full textInternational audienceIn this article, we study the weak uniqueness and the regularity of the weak solutions of a fluid-structure interaction system. More precisely, we consider the motion of a rigid ball in a viscous incompressible fluid and we assume that the fluid-rigid body system fills the entire space . We prove that the corresponding weak solutions that additionally satisfy a classical Prodi-Serrin condition, including a critical one, are unique. We also show that the weak solutions are regular under the Prodi-Serrin conditions, with a smallness condition in the critical case
Three critical points theorem and its application to quasilinear elliptic equations
No full textAbstractIn this paper, we prove a Pucci–Serrin type three critical points theorem for continuous functionals and study its application to quasilinear elliptic equations with natural growth
On the interior regularity criteria and the number of singular points to the Navier-Stokes equations
No full textWe establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin's criterion and Escauriza-Seregin-verak's criterion. We also show that if a weak solution u satisfies parallel to u(.,t)parallel to L-P <= C(-t)((3-P)/2p) for some 3 < p < a, then the number of singular points is finite.MathematicsSCI(E)[email protected]; [email protected]
On positive solutions of quasi-linear elliptic equations involving critical Sobolev growth
No full textAbstractWe study the boundary value problem of the quasi-linear elliptic equationdiv(|∇u|m−2∇u)+f(x,u,∇u)=0in Ω,u=0on ∂Ω, where Ω⊂Rn (n⩾2) is a connected smooth domain, and the exponent m∈(1,n) is a positive number. Under appropriate conditions on the function f, a variety of results on existence and non-existence of positive solutions have been established. This paper is a continuation of an earlier work Zou (2008) [18] of the author and, in particular, extends earlier results of Brezis and Nirenberg (1983) [3] for the semi-linear case of m=2, and of Pucci and Serrin (1986) [12] for the quasi-linear case of m≠2
Remarks about the mean value property and some weighted Poincar\'e-type inequalities
No full textWe start providing a quantitative stability theorem for the rigidity of an
overdetermined problem involving harmonic functions in a punctured domain. Our
approach is inspired by and based on the proof of rigidity established by
Enciso and Peralta-Salas, and reveals essential differences with respect to the
stability results obtained in the literature for the classical overdetermined
Serrin problem.
Secondly, we provide new weighted Poincar\'e-type inequalities for vector
fields. These are crucial tools for the study of the quantitative stability
issue initiated by the author concerning a class of rigidity results involving
mixed boundary value problems.
Finally, we provide a mean value-type property and an associated weighted
Poincar\'e-type inequality for harmonic functions in cones. A duality relation
between this new mean value property and a partially overdetermined boundary
value problem is discussed, providing an extension of a classical result due to
Payne and Schaefer
The set of positive solutions of semilinear equations in large balls
No full textSynopsisThe exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).</jats:p
Nonlinear parabolic stochastic evolution equations in critical spaces part II: Blow-up criteria and instantaneous regularization
No full textThis paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical L2-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.Analysi
Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity
No full textWe are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebraic time-decay estimates in critical Besov spaces (and even exponential decay for the high frequencies) for any derivatives of the solution. Our approach is partly inspired by the work of Bae, Biswas & Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires our establishing new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, this is the first work pointing out Gevrey analyticity for a model of compressible fluids
Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity
No full textInternational audienceWe are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebraic time-decay estimates in critical Besov spaces (and even exponential decay for the high frequencies) for any derivatives of the solution. Our approach is partly inspired by the work of Bae, Biswas & Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires our establishing new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, this is the first work pointing out Gevrey analyticity for a model of compressible fluids
Spatial Behavior Backward in Time
No full textIn this entry, the spatial behavior of solutions for the backward-in-time problem of the linear theory of thermoelasticity is studied. In this type of problem final data are assigned, usually at the time t=0, instead of initial data, and then we are interested in extrapolating to previous times. We associate with a solution of the considered problem an appropriate time-weighted volume measure, for which we get a spatial estimate describing a spatial exponential decay of the solution. The backward-in-time problems have been initially considered by Serrin [1] who established uniqueness results for the Navier–Stokes equations. Explicit uniqueness and stability criteria for classical Navier–Stokes equations backward in time have been further established by Knops and Payne [2] and Galdi and Straughan [3] (see also Payne and Straughan [4] for a class of improperly posed problems for parabolic partial differential equations). Such backward-in-time problems have been considered also by Ames and Payne [5] in order to obtain stabilizing criteria for solutions of the boundary-final value problem. It is well known that this type of problem is ill posed. In [6], Ciarletta established uniqueness and continuous dependence results upon mild require- ments concerning the thermoelastic coefficients; in particular the author considers hypotheses not real- istic from the physical point of view, such as a positive semidefinite elasticity tensor or a nonpositive heat capacity. Moreover, introducing an appropriate time-weighted volume measure, Ciarletta and Chiria [7] established the spatial estimate describing the spatial exponential decay of the thermoelastic process backward in time
