The Experts below are selected from a list of 7950 Experts worldwide ranked by ideXlab platform
Alberto Tesei - One of the best experts on this subject based on the ideXlab platform.
-
discontinuous solutions of hamilton jacobi equations versus Radon Measure valued solutions of scalar conservation laws disappearance of singularities
2021Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto TeseiAbstract:Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation $$U_{t}+H(U_x)=0$$ and signed Radon Measure valued entropy solutions of the conservation law $$u_{t}+[H(u)]_x=0$$ . After having proved a precise statement of the formal relation $$U_x=u$$ , we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton–Jacobi equation and signed singular Measures in case of the conservation law.
-
a uniqueness criterion for Measure valued solutions of scalar hyperbolic conservation laws
2019Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto TeseiAbstract:We prove existence and uniqueness of Radon Measure-valued solutions of the Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0 &\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ a positive Radon Measure whose singular part is a finite superposition of Dirac masses, and $\varphi\in C^2([0,\infty))$ is bounded. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.
-
Radon Measure valued solutions of first order hyperbolic conservation laws
2018Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto TeseiAbstract:We study nonnegative solutions of the Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0&\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ is a Radon Measure and $\varphi:[0,\infty)\mapsto \mathbb{R}$ is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon Measures. Under some additional conditions on $\varphi$, we prove their uniqueness if the singular part of $u_0$ is a finite superposition of Dirac masses. In terms of the behaviour of $\varphi$ at infinity we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive {\em waiting time} (in the linear case $\varphi(u)=u$ this happens for all times). In the latter case we describe the evolution of the singular parts.
-
on a class of forward backward parabolic equations existence of solutions
2017Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Alberto TeseiAbstract:Abstract We study the initial–boundary value problem u t = [ φ ( u ) ] x x + ϵ [ ψ ( u ) ] t x x in Ω × ( 0 , T ) φ ( u ) + ϵ [ ψ ( u ) ] t = 0 in ∂ Ω × ( 0 , T ) u = u 0 in Ω × { 0 } , where Ω is an interval and u 0 is a nonnegative Radon Measure on Ω . The map φ is increasing in ( 0 , α ) and decreasing in ( α , ∞ ) for some α > 0 , and satisfies φ ( 0 ) = φ ( ∞ ) = 0 . The regularizing map ψ is increasing and bounded. We prove existence of suitably defined nonnegative Radon Measure-valued solutions. The solution class is natural since smooth initial data may generate solutions which become Measure-valued after finite time.
-
on a class of forward backward parabolic equations properties of solutions
2017Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Alberto TeseiAbstract:We study the equation $u_t=[\varphi(u)]_{xx}+\varepsilon [\psi(u)]_{txx}$ with suitable boundary conditions and a nonnegative Radon Measure as initial datum. Here $\varphi(0)=\varphi(\infty)=0$, $\...
Flavia Smarrazzo - One of the best experts on this subject based on the ideXlab platform.
-
discontinuous solutions of hamilton jacobi equations versus Radon Measure valued solutions of scalar conservation laws disappearance of singularities
2021Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto TeseiAbstract:Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation $$U_{t}+H(U_x)=0$$ and signed Radon Measure valued entropy solutions of the conservation law $$u_{t}+[H(u)]_x=0$$ . After having proved a precise statement of the formal relation $$U_x=u$$ , we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton–Jacobi equation and signed singular Measures in case of the conservation law.
-
a uniqueness criterion for Measure valued solutions of scalar hyperbolic conservation laws
2019Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto TeseiAbstract:We prove existence and uniqueness of Radon Measure-valued solutions of the Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0 &\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ a positive Radon Measure whose singular part is a finite superposition of Dirac masses, and $\varphi\in C^2([0,\infty))$ is bounded. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.
-
Radon Measure valued solutions of first order hyperbolic conservation laws
2018Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto TeseiAbstract:We study nonnegative solutions of the Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0&\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ is a Radon Measure and $\varphi:[0,\infty)\mapsto \mathbb{R}$ is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon Measures. Under some additional conditions on $\varphi$, we prove their uniqueness if the singular part of $u_0$ is a finite superposition of Dirac masses. In terms of the behaviour of $\varphi$ at infinity we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive {\em waiting time} (in the linear case $\varphi(u)=u$ this happens for all times). In the latter case we describe the evolution of the singular parts.
-
on a class of forward backward parabolic equations existence of solutions
2017Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Alberto TeseiAbstract:Abstract We study the initial–boundary value problem u t = [ φ ( u ) ] x x + ϵ [ ψ ( u ) ] t x x in Ω × ( 0 , T ) φ ( u ) + ϵ [ ψ ( u ) ] t = 0 in ∂ Ω × ( 0 , T ) u = u 0 in Ω × { 0 } , where Ω is an interval and u 0 is a nonnegative Radon Measure on Ω . The map φ is increasing in ( 0 , α ) and decreasing in ( α , ∞ ) for some α > 0 , and satisfies φ ( 0 ) = φ ( ∞ ) = 0 . The regularizing map ψ is increasing and bounded. We prove existence of suitably defined nonnegative Radon Measure-valued solutions. The solution class is natural since smooth initial data may generate solutions which become Measure-valued after finite time.
-
on a class of forward backward parabolic equations properties of solutions
2017Co-Authors: Michiel Bertsch, Flavia Smarrazzo, Alberto TeseiAbstract:We study the equation $u_t=[\varphi(u)]_{xx}+\varepsilon [\psi(u)]_{txx}$ with suitable boundary conditions and a nonnegative Radon Measure as initial datum. Here $\varphi(0)=\varphi(\infty)=0$, $\...
Tesei A. - One of the best experts on this subject based on the ideXlab platform.
-
Discontinuous solutions of Hamilton-Jacobi equations versus Radon Measure-valued solutions of scalar conservation laws: Disappearance of singularities
2020Co-Authors: Bertsch M., Smarrazzo F., Terracina A., Tesei A.Abstract:Let $H$ be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation $U_{t}+H(U_x)=0$ and signed Radon Measure valued entropy solutions of the conservation law $u_{t}+[H(u)]_x=0$. After having proved a precise statement of the formal relation $U_x=u$, we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular Measures in case of the conservation law.Comment: 33 page
-
Radon Measure-valued solutions of first order scalar conservation laws
2020Co-Authors: Bertsch M., Smarrazzo F., Terracina A., Tesei A.Abstract:We study nonnegative solutions of the Cauchy problempartial derivative(t)u + partial derivative(x)[phi(u)] = 0 in R x (0, T),u = u(0) >= 0 in R x 0,where u(0) is a Radon Measure and phi [0, infinity) bar right arrow R is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon Measures. Under some additional conditions on phi, we prove their uniqueness if the singular part of u(0) is a finite superposition of Dirac masses. Regarding the behavior of phi at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case phi(u) = u this happens for all times). In the latter case, we describe the evolution of the singular parts
-
Signed Radon Measure-valued solutions of flux saturated scalar conservation laws
2019Co-Authors: Bertsch M., Smarrazzo F., Terracina A., Tesei A.Abstract:We prove existence and uniqueness for a class of signed Radon Measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness
-
A uniqueness criterion for Measure-valued solutions of scalar hyperbolic conservation laws,
2019Co-Authors: Bertsch M., Smarrazzo F., Terracina A., Tesei A.Abstract:We prove existence and uniqueness of Radon Measure-valued solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, the initial data being a finite superposition of Dirac masses and the flux being Lipschitz continuous, bounded and sufficiently smooth. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness
Maria Michaela Porzio - One of the best experts on this subject based on the ideXlab platform.
-
Radon Measure valued solutions for some quasilinear degenerate elliptic equations
2015Co-Authors: Maria Michaela Porzio, Flavia SmarrazzoAbstract:We prove the existence of suitably defined weak Radon Measure-valued solutions for a class of quasilinear elliptic equations. Moreover, we prove uniqueness results introducing the notion of “weak entropy solutions.”
-
Radon Measure valued solutions of nonlinear strongly degenerate parabolic equations
2014Co-Authors: Maria Michaela Porzio, Flavia Smarrazzo, Alberto TeseiAbstract:We prove the existence of suitably defined weak Radon Measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: $$(i)$$ the concentrated part of the solution with respect to the Newtonian capacity is constant; $$(ii)$$ the total variation of the singular part of the solution (with respect to the Lebesgue Measure) is nonincreasing in time. Conditions under which Radon Measure-valued solutions of problem $$(P)$$ are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given.
-
Radon Measure valued solutions for a class of quasilinear parabolic equations
2013Co-Authors: Maria Michaela Porzio, Flavia Smarrazzo, Alberto TeseiAbstract:Initial value problems for quasilinear parabolic equations having Radon Measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon Measures of the initial data. We call such solutions Measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of Measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue Measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue Measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the Measure-valued solution of problem (P) is, in fact, function-valued.
Debajyoti Choudhuri - One of the best experts on this subject based on the ideXlab platform.
-
a critical fractional choquard problem involving a singular nonlinearity and a Radon Measure
2021Co-Authors: Akasmika Panda, Debajyoti Choudhuri, Kamel SaoudiAbstract:This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1 $$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$ Here, $$\Omega $$ is a bounded domain of $$\mathbb {R}^N$$ , $$s\in (0,1)$$ , $$\alpha ,\lambda $$ and $$\beta $$ are positive real parameters, $$N>2s$$ , $$\gamma \in (0,1)$$ , $$0Sobolev inequality and $$\mu $$ is a bounded Radon Measure in $$\Omega $$ .
-
singular fractional choquard equation with a critical nonlinearity and a Radon Measure
2020Co-Authors: Akasmika Panda, Debajyoti Choudhuri, Kamel SaoudiAbstract:This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
-
existence of solution for a system involving fractional laplacians and a Radon Measure
2019Co-Authors: Amita Soni, Debajyoti ChoudhuriAbstract:An existence of a nontrivial solution in some `weaker' sense of the following system of equations \begin{align*} (-\Delta)^{s}u+l(x)\phi u+w(x)|u|^{k-1}u&=\mu~\text{in}~\Omega\nonumber\\ (-\Delta)^{s}\phi&= l(x)u^2~\text{in}~\Omega\nonumber\\ u=\phi&=0 ~\text{in}~\mathbb{R}^N\setminus\Omega \end{align*} has been proved. Here $s \in (0,1)$, $l,w$ are bounded nonnegative functions in $\Omega$, $\mu$ is a Radon Measure and $k > 1$ belongs to a certain range.
-
elliptic partial differential equation involving a singularity and a Radon Measure
2018Co-Authors: Akasmika Panda, Sekhar Ghosh, Debajyoti ChoudhuriAbstract:The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon Measure μ as its nonhomogenous term which is given as −Δu = f(x)h(u) + μ in Ω, u = 0 on ∂Ω, u > 0 on Ω, where Ω is a bounded domain of R N , f is a nonnegative function over Ω.
-
existence of solution of the p x laplacian problem involving critical exponent and Radon Measure
2018Co-Authors: Amita Soni, Debajyoti ChoudhuriAbstract:In this paper we are proving the existence of a nontrivial solution of the ${p}(x)$- Laplacian equation with Dirichlet boundary condition. We will use the variational method and concentration compactness principle involving positive Radon Measure $\mu$. \begin{align*} \begin{split} -\Delta_{p(x)}u & = |u|^{q(x)-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\, \mbox{on}\,\, \partial\Omega, \end{split} \end{align*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $\mu > 0$ and $1 < p^{-}:=\underset{x\in \Omega}{\text{inf}}\;p(x) \leq p^{+}:= \underset{x\in \Omega}{\text{sup}}\;p(x) < q^{-}:=\underset{x\in \Omega}{\text{inf}}\;q(x)\leq q(x) \leq p^{\ast}(x) < N$. The function $f$ satisfies certain conditions. Here, $q^{\prime}(x)=\frac{q(x)}{q(x)-1}$ is the conjugate of $q(x)$ and $p^{\ast}(x)=\frac{Np(x)}{N-p(x)}$ is the Sobolev conjugate of $p(x)$.